We claim that, for every sufficiently high snc model ${\mathcal{X}}^{\prime}$ proper over ${\mathcal{X}}$, $v^{\prime}:=r_{{\mathcal{X}}^{\prime}}(v)$ and $v$ have the same center on ${\mathcal{X}}$. Indeed, the center of $v$ on ${\mathcal{X}}^{\prime}$ is a specialization of that of $r_{{\mathcal{X}}^{\prime}}(v)$, and hence $c_{\mathcal{X}}(v)\in\overline{c_{\mathcal{X}}(r_{{\mathcal{X}}^{\prime}}(v))}$. On the other hand, we have $\lim_{{\mathcal{X}}^{\prime}}r_{{\mathcal{X}}^{\prime}}(v)=v$. Since $c_{\mathcal{X}}\colon{\mathcal{X}}^{\mathrm{an}}\to{\mathcal{X}}_{0}$ is anticontinuous, $c_{\mathcal{X}}^{-1}(\overline{\{c_{\mathcal{X}}(v)\}})$ is open, and hence contains $v^{\prime}:=r_{{\mathcal{X}}^{\prime}}(v)$ for some snc model ${\mathcal{X}}^{\prime}$ proper over ${\mathcal{X}}$. As a result, $c_{\mathcal{X}}(v^{\prime})$ is a specialization of $c_{\mathcal{X}}(v)$, and the claim follows.

By (5.3), we have $A_{\mathcal{X}}(v^{\prime})=0$, and it is thus enough to prove the result for $v^{\prime}\in\operatorname{Sk}({\mathcal{X}}^{\prime})$. If $\sigma$ is the unique face of $\Delta({\mathcal{X}}^{\prime})$ containing $v^{\prime}$ in its interior, then $A_{\mathcal{X}}\equiv 0$ on $\sigma$, since $A_{\mathcal{X}}$ is non-negative and affine on $\sigma$. For any divisorial point $w$ in the relative interior of $\sigma$, we thus have $A_{\mathcal{X}}(w)=0$ and $c_{{\mathcal{X}}^{\prime}}(v^{\prime})=c_{{\mathcal{X}}^{\prime}}(w)$, which shows that $c_{\mathcal{X}}(v^{\prime})=c_{\mathcal{X}}(w)$ is an lc center. ∎