\times 10^{-4} \times 
\left(
 \frac{m_{A}^{}}{350 {\rm [GeV]}}
\right)^{4}
\left(
\frac{30}{\tan\beta}
\right)^{6}.
\label{eq:kappa32sqmax}\end{aligned}$$ It can be easily seen that the bound $(|\kappa_{32}^{\text{max}}|^{2})_{\tau\rightarrow \mu\eta}$ is rapidly relaxed in the region with small $\tan\beta$ and large $m_{A}^{}$. In a similar way to Eq. (\[eq:bound-from-tmeta\]), the maximal allowed value $\left( |\kappa_{3i}^{\text{max}}|^{2} \right)_{\text{mode}}$ can be calculated for each mode. The combined upper limit $|\kappa^{\text{max}}_{3i}|^{2}$ is then given by $$\begin{aligned}
\left|\kappa^{\text{max}}_{3i}\right|^{2}
 \equiv
 {\rm MIN}
 \left\{
  \left(
  \left|
   \kappa^{\text{max}}_{3i}
  \right|^{2}
  \right)_{\tau\rightarrow\ell\eta},
  \left(
  \left|
   \kappa^{\text{max}}_{3i}
  \right|^{2}
  \right)_{\tau \rightarrow \ell \mu^{+}\mu^{-}},
  \left(
  \left|
   \kappa^{\text{max}}_{3i}
  \right|^{2}
  \right)_{\tau \rightarrow \ell K^{+}K^{-}},
  \left(
  \left|
   \kappa^{\text{max}}_{3i}
  \right|^{2}
  \right)_{\tau \rightarrow \ell \gamma}, \cdot\cdot\cdot
 \right\}.
\label{eq:kappaSq-max}\end{aligned}$$ As shown below, $\tau^{}\rightarrow \ell^{}_i\eta$ and $\tau^{}\rightarrow \ell^{}_i\gamma$ give the strongest upper limits on $|\kappa_{3i}|^{2}$ in a wide range of the parameter space. In addition, in some parameter regions $\tau^{}\rightarrow \ell^{}_iK^{+}K^{-}$ and $\tau^{}\rightarrow \ell^{}_i\mu^{+}\mu^{-}$ can also give similar limits on $|\kappa_{3i}|^{2}$ to those from the above two processes.

(7.8,6) (0,1)[![ Contours of $|\kappa^{\text{max}}_{32}|^{2}$, the possible maximal value of $|\kappa_{32}|^{2}$ from the rare tau decay results, are shown (a) in the $\tan\beta$-$m_{A}$ plane and (b) in the $\tan\beta$-$\sin^{2}(\alpha-\beta)$ plane. The parameters are taken to be (a) $m_{h}^{} =120$ GeV, $m_{H}^{} = m_{H^{\pm}} = 350$ GeV and $\sin(\alpha -\beta) = -0.9999$, 