(Regularity of real MA solution) Inside $\partial\Delta_{\lambda}^{\vee}\setminus Sing$, let $\mathcal{R}$ be the set of strictly convex points of $u_{\infty}$, then $u_{\infty}\in C^{\infty}_{loc}(\mathcal{R})$, and the complement of $\mathcal{R}$ is a closed subset of Hausdorff $(n-1)$-measure zero. In particular $\mathcal{R}$ is path connected, and is open and dense in $\partial\Delta_{\lambda}^{\vee}\setminus Sing$.