of them. When all the masses of Higgs bosons are large, these decay processes decouple by a factor of $1/m_{\rm Higgs}^{4}$. Although these branching ratios are complicated functions of the mixing angles, each of them can be simply expressed to be proportional to $\tan^{6} \beta$ for $\tan\beta \gg 1$ in the SM-like region ($\sin(\alpha-\beta) \sim -1$ [@GH; @smlike1; @smlike2]). This $\tan^{6} \beta$ dependence is a common feature of the tau-associated LFV processes with the Higgs-mediated 4-Fermi interactions.

The experimental upper limit on $|\kappa_{3i}|^{2} (\equiv |\kappa_{3i}^L|^{2}+|\kappa_{i3}^R|^{2})$ can be obtained by using the experimental results given in Table \[Tab:tau-bound\] and analytic expressions of the decay branching ratios for rare tau LFV decay processes in Ref. [@kot]. For description, let us consider the bound from the $\tau\rightarrow \mu\eta$ results[@Sher-tmeta]; $$\begin{aligned}
&|\kappa_{32}|^{2} \leq
 \left( 
 \left|
  \kappa^{\text{max}}_{32}
 \right|^{2}
 \right)_{\tau^{}\rightarrow \mu^{}\eta}
 \equiv
 \frac{256 \pi {\rm Br}(\tau^{} \rightarrow \mu^{} \eta)_{\rm exp}  m_{A}^{4}}
 {9 G_{F}^{2} m_{\tau}^{3} m_{\eta}^{4} F_{\eta}^{2} \tau_{\tau} 
  \left(
   1- \frac{m_{\eta}^{2}}{m_{\tau}^{2}}
  \right)^{2}}
 \frac{\cos^{6}\beta}{\sin^{2}\beta},
\label{eq:bound-from-tmeta}\end{aligned}$$ where ${\rm Br}(\tau^{} \rightarrow \mu^{} \eta)_{\rm exp}$ is the experimental upper limit on the branching ratio of $\tau^{} \rightarrow
\mu^{} \eta$ in Table \[Tab:tau-bound\]. In particular, for $\tan\beta\gg1$, the right-hand-side can be expressed by $$\begin{aligned}
 \left( 
 \left|
  \kappa^{\text{max}}_{32}
 \right|^{2}
 \right)_{\tau^{}\rightarrow \mu^{}\eta}
\simeq
2.3 