results of the tau LFV decays do not give any substantial upper limit on $|\kappa_{32}|^2$ except for high $\tan\beta$ region ($\tan\beta \gtrsim 30$).

Finally, we comment on the bound on $|\kappa_{31}|^2$, the LFV parameters for $\tau$-$e$ mixing. Similar to $\tau$-$\mu$ mixing, we can discuss $|\kappa_{31}^{\rm max}|^2$ comparing the data of $\tau \to e \eta$, $\tau \to e \mu^+\mu^-$, $\tau \to e K^+K^-$ and $\tau \to e \gamma$ listed in Table \[Tab:tau-bound\] with the formulas given in Ref. [@kot]. These formulas for $\tau$-$e$ mixing are common with $\tau$-$\mu$ mixing except for the factor of $|\kappa_{3i}|^2$, so that difference in contours of $|\kappa_{31}^{\rm max}|^2$ from those of $|\kappa_{32}^{\rm max}|^2$ only comes from that in the data. In Table \[Tab:tau-bound\], the experimental limit for the branching ratio of $\tau \to e \eta$ is about 1.5 times weaker than that of $\tau \to \mu \eta$, while that of $\tau \to e \gamma$ is 5.7 times relaxed as compared to that of $\tau \to \mu \gamma$. Moreover, the upper limit on ${\rm Br}(\tau^- \to e^-K^+K^-)$ is 1.8 times stronger than that on ${\rm Br}(\tau^- \to \mu^-K^+K^-)$. We have numerically confirmed that there are some regions where $\tau^- \to e^-K^+K^-$ can give the most stringent bound 