Notice there is a gap between the above results and the Gromov-Hausdorff convergence to the real MA metric on $Sk(X)$ defined by the Hessian of $\phi_{0}$, because the $n$-dimensional open faces are disconnected, and therefore we cannot access the distance of two points on different faces. One needs further information on the $(n-1)$-dimensional faces of $Sk(X)$.