Since the relative log canonical divisor $K^{\mathrm{log}}_{{\mathcal{X}}/S}$ and ${\mathcal{L}}$ are both models of $K_{X}$, $D:=K^{\mathrm{log}}_{{\mathcal{X}}/S}-{\mathcal{L}}$ is a ${\mathbb{Q}}$-Cartier divisor supported on ${\mathcal{X}}_{0}$. The corresponding model function $\phi_{D}$ satisfies $\kappa=A_{\mathcal{X}}+\phi_{D}$, which shows that $\kappa|_{\operatorname{Sk}({\mathcal{X}})}=\phi_{D}|_{\operatorname{Sk}({\mathcal{X}})}$ is affine on each face of $\Delta({\mathcal{X}})$. Now pick $v\in\operatorname{Sk}(\psi)$. By (5.1), we get

$$\kappa(v)=A_{\mathcal{X}}(v)+\phi_{D}(v)\geq\inf\phi_{D}=\min_{i}\phi_{D}(v_{i})=\min_{i}(A_{\mathcal{X}}+\phi_{D})(v_{i})\geq\inf_{X^{\mathrm{an}}}\kappa.$$

It follows that $A_{\mathcal{X}}(v)=0$, and hence $v\in\operatorname{Sk}({\mathcal{X}})$, by Proposition 5.6. ∎