dark shaded area for the criterion $\xi=1$ and by the light one for $\xi=1/2$. In Fig. \[Fig:allowed-kappa\]-(a), parameters of the Higgs sector are taken to be $m_{h}^{} =120$ GeV, $m_{H}^{} = m_{H^{\pm}}^{} = 350$ GeV and $\sin(\alpha -\beta) = -0.9999$. In Fig. \[Fig:allowed-kappa\]-(b), those are $m_{h}^{} =120$ GeV and $m_{H}^{} = m_{A}^{} = m_{H^{\pm}}^{} = 350$ GeV. The value of $|\kappa^{\text{max}}_{32}|^{2}$ is independent of $M$, the soft-breaking scale of the discrete symmetry. On the other hand, theoretical bounds from vacuum stability and perturbative unitarity are sensitive to $M$. Therefore, we evaluate such a theoretical allowed region by scanning $M$ to be from 0 to 1000 GeV. We also take into account the constraint from the $\rho$ parameter measurement and the $b\rightarrow s \gamma$ result by taking $\sin(\alpha-\beta) \simeq -1$ and $m_{H}^{}=m_{H^{\pm}}^{}$ with $m_{H^{\pm}}^{}\gtrsim 350$ GeV for Fig \[Fig:allowed-kappa\]-(a), and $m_{A}^{}=m_{H^{\pm}}^{}$ with $m_{H^{\pm}}^{}\gtrsim350$ GeV for Fig \[Fig:allowed-kappa\]-(b). From the both figures, it is easily found that the value of $|\kappa_{32}^{\text{max}}|^{2}$ can extensively be larger for smaller $\tan\beta$ in the allowed region under the theoretical constraints. For $\tan\beta \lesssim 10$ $(30)$, $|\kappa_{32}^{\text{max}}|^{2}$ can be $\mathcal{O}(0.1)$ ($\mathcal{O}(10^{-4})$). Among the rare tau decay processes, $\tau \rightarrow \mu\eta$ and $\tau \rightarrow \mu \gamma$ provide the 