(Star type regions on $X_{s}$) On the region $U^{s}_{w}\subset X_{s}$, recall the coodinates $z^{m_{i}}$ and regard $x^{m_{i}}=s^{-1}\log|z^{m_{i}}|$ as local coordinates also on $U^{\infty}_{w}\cap\partial\Delta_{\lambda}^{\vee}$. Let $U^{s,*}_{w}\subset U^{s}_{w,\delta}$ be the subset where the $x^{m_{i}}$ coordinates correspond to points in $\text{Star}(w)\subset\partial\Delta_{\lambda}^{\vee}$. The tropical analogue of $U^{s,*}_{w}$ is $(\text{Star}(w)+\mathbb{R}_{\geq 0}w)\cap A_{\lambda}^{\infty}$.