For $m=1$, consider a $0$-dimensional face $\Delta_{J}$ of $Sk(X)$, such that there is a unique $1$-dimensional face of $\Delta_{\mathcal{X}}$ containing $\Delta_{J}$, and it lies on $Sk(X)$. The simplest possibility $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})=0$ imposes a Neumann-like boundary condition on $\phi_{0}$. This example may describe the Tian-Yau region in [25][42].