most stringent constraints on $|\kappa_{32}^{}|^{2}$. While $\tau \rightarrow \mu \eta$ is mediated only by $A$, $\tau \rightarrow \mu\gamma$ depends on the masses of $h$, $H$, $A$ and $H^\pm$. For $\sin^{2}(\alpha-\beta) \sim 1$ and $m_{A}^{} \sim m_{H}^{}$, the branching ratio of $\tau\to\mu\gamma$ is suppressed because of the cancellation between the one-loop diagrams of $A$ and $H$. Therefore, $|\kappa_{32}|^2$ is bounded most strongly by the $\tau \rightarrow \mu \eta$ result for this case[^4]. When $m_A$ differs from $m_H$ or when $\sin^2(\alpha-\beta)$ is to some extent smaller than unity, the one-loop induced $\tau\rightarrow \mu\gamma$ process becomes important, and gives the most stringent bound on $|\kappa_{32}|^2$ of all the rare tau decay processes.

The value of $|\kappa^{\text{max}}_{32}|^{2}$ can be much larger than 100 in a wide range of the parameter region. One might think that such large values of $|\kappa_{3i}|$ cannot be consistent with the unitarity argument for the LFV Yukawa couplings. However, it should be emphasized that the above figures show the contour plots for $|\kappa_{32}^{\text{max}}|^{2}$ under the rare tau decay results, and not for $|\kappa_{32}|^{2}$. The region of $|\kappa^{\text{max}}_{32}|^{2} \gtrsim 1$ should be taken as the area where $|\kappa_{32}|^{2} $ can be as large as $\mathcal{O}(10^{-2}\text{-}10^{-4})$ easily. It is concluded that current 