$M_{\tau}$ distribution in signal region, (OS$\mu$, Belle II)

$M_{\tau}$ distribution in signal region, (SS$e$, Belle)

$M_{\tau}$ distribution in signal region, (SS$e$, Belle II)

$M_{\tau}$ distribution in signal region, (SS$\mu$, Belle)

$M_{\tau}$ distribution in signal region, (SS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$e$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$e$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$e$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$e$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$e$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$e$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$\mu$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$\mu$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$e$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$e$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$\mu$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$\mu$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$e$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$e$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$\mu$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$\mu$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$e$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$e$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$\mu$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$\mu$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (OS$e$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (OS$e$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (OS$\mu$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (OS$\mu$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (SS$e$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (SS$e$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (SS$\mu$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (SS$\mu$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$e$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$e$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$e$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$e$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle II)

The measured differential branching fractions as a function of the invariant hadronic mass squared of the $X_u$ system ($M_X^2$).

The measured differential branching fractions as a function of the light-cone momenta of the hadronic $X_u$ system [$P_+ = (E_X^B - |old{p}_X^B|)$].

The measured differential branching fractions as a function of the light-cone momenta of the hadronic $X_u$ system [$P_- = (E_X^B + |old{p}_X^B|)$].

The background-subtracted yields as a function of $E_\ell^B$.

The background-subtracted yields as a function of $q^2$.

The background-subtracted yields as a function of $M_X$.

The background-subtracted yields as a function of $M_X^2$.

The background-subtracted yields as a function of $P_+$. [$P_+ = (E_X^B - |P_X^B|)$]

The background-subtracted yields as a function of $P_-$. [$P_- = (E_X^B + |P_X^B|)$].

The full experimental correlation matrix for the background-subtracted spectrum of the lepton energy in the $B$ rest frame ($E_\ell^B$).

The full experimental correlation matrix for the background-subtracted spectrum of $q^2$).

The full experimental correlation matrix for the background-subtracted spectrum of $M_X$.

The full experimental correlation matrix for the background-subtracted spectrum of $M_X^2$.

The full experimental correlation matrix for the background-subtracted spectrum of $P_+$.

The full experimental correlation matrix for the background-subtracted spectrum of $P_-$.

Migration matrix of $E_\ell^B$ [in %].

Migration matrix of $q^2$ [in %].

Migration matrix of $M_X$ [in %].

Migration matrix of $M_X^2$ [in %].

Migration matrix of $P_+$ [in %].

Migration matrix of $P_-$ [in %].

The total correction factors and phase space acceptance as a function of the invariant hadronic mass of the $X_u$ system ($M_X$).

The total correction factors and phase space acceptance as a function of the invariant hadronic mass squared of the $X_u$ system ($M_X^2$).

The total correction factors and phase space acceptance as a function of the lepton energy in the $B$ rest frame ($E_\ell^B$).

The total correction factors and phase space acceptance as a function of the four-momentum-transfer squared of the $B$ to the $X_u$ system ($q^2$).

The total correction factors and phase space acceptance as a function of $P_+$.

The total correction factors and phase space acceptance as a function of $P_-$.

The statistical correlations of the differential branching fractions are shown. The matrix elements are ordered for $M_{X}$, $M_{X}^{2}$, $q^{2}$, $E_{\ell}^{B}$, $P_{+}$ and $P_{-}$.

The full experimental (statistical and systematical) correlations of the differential branching fractions are shown. The matrix elements are ordered for $M_{X}$, $M_{X}^{2}$, $q^{2}$, $E_{\ell}^{B}$, $P_{+}$ and $P_{-}$.

The total partial branching fraction with $E_{\ell}^{B} > 1$ GeV. The order of entries is: the 2D fit result of Phys. Rev. D 104, 012008 (2021), the results calcuated with the differential branching fractions of $M_{X}$, $M_{X}^{2}$, $q^{2}$, $E_{\ell}^{B}$, $P_{+}$ and $P_{-}$

The full experimental correlations between the total partial branching fractions from summing the individual bins are shown. The matrix elements are saved in the order of $M_{X}$, $M_{X}^{2}$, $q^{2}$, $E_{\ell}^{B}$, $P_{+}$ and $P_{-}$.

The first moment of the measured differential branching fraction of the lepton energy in the $B$ rest frame ($E_\ell^B$).

The second moment of the measured differential branching fraction of the lepton energy in the $B$ rest frame ($E_\ell^B$).

The third moment of the measured differential branching fraction of the lepton energy in the $B$ rest frame ($E_\ell^B$).

The first moment of the measured differential branching fraction of $q^2$.

The second moment of the measured differential branching fraction of $q^2$.

The third moment of the measured differential branching fraction of $q^2$.

The first moment of the measured differential branching fraction of $M_{X}$.

The second moment of the measured differential branching fraction of $M_{X}$.

The third moment of the measured differential branching fraction of $M_{X}$.

The first moment of the measured differential branching fraction of $P_{+}$.

The second moment of the measured differential branching fraction of $P_{+}$.

The third moment of the measured differential branching fraction of $P_{+}$.

The first moment of the measured differential branching fraction of $P_{-}$.

The second moment of the measured differential branching fraction of $P_{-}$.

The third moment of the measured differential branching fraction of $P_{-}$.

The matrix elements are saved in the order of $M_{X}$, $q^{2}$, $E_{\ell}^{B}$, $P_{+}$ and $P_{-}$. For each variable, the entries is ordered for the first, second and third moments.

The unfolded differential rate as a function of $\chi$.

The correlation matrix of the unfolded differential rates. The full error covariance can be obtained by combining the quoted error in the other tables with these values. The ordering of the correlations is {$w$, $\cos\theta_\nu$, $\cos\theta_\ell$, $\chi$}, where each variable has 10 bins.