Let $\mathbb{C}_{K}$ be the completion of an algebraic closure of $K$. Then there are exactly $\Deg(S)$ points in the special fibre of $\mathfrak{X}_{\mathbb{C}_{K}}$ mapping to $S$, hence there are precisely $\Deg(S)$ open faces in the skeleton associated to $\mathfrak{X}_{\mathbb{C}_{K}}$ lying over $\tau$. As the base change induces an isomorphism of each of these faces with $\tau$, we have $\iota_{\ast}\MA(\iota^{\ast}h)=\Deg(S)\MA(h)$. Using this and the invariance of the non-archimedean Monge-Ampère measure under base change we may assume $K=\mathbb{C}_{K}$. As in the proof of Proposition 5.9 we choose a strongly nondegenerate strictly polystable formal scheme $\mathfrak{X}^{\prime}$ and a surjective étale morphism $\varphi:\mathfrak{X}^{\prime}\rightarrow\mathfrak{X}$. Let $\tau^{\prime}$ be an open face of the skeleton associated to $\mathfrak{X}^{\prime}$ lying over $\tau$. As we have seen, $\varphi$ induces an isomorphism $p_{\mathfrak{X}^{\prime}}^{-1}(\tau^{\prime})\tilde{\rightarrow}p_{\mathfrak{X}}^{-1}(\tau)$. As in the proof of Corollary 5.7 there is a sequence of rational piecewise affine linear convex functions $(h^{\prime}_{i})_{i\in\mathbb{N}}$ on $\tau^{\prime}$ converging locally uniformly to $h\circ\varphi^{\textup{an}}$. Let $h_{i}$ be the piecewise affine linear functions on $\tau$ such that $h_{i}\circ\varphi^{\textup{an}}=h^{\prime}_{i}$. By Proposition 5.9 the metrics induced by the $h_{i}$ are semipositive piecewise $\mathbb{Q}$-linear metrics on $p_{\mathfrak{X}}^{-1}(\tau)$ which implies that the metric induced by $h$ is locally semipositive. As the restriction of $\varphi$ to $p_{\mathfrak{X}^{\prime}}^{-1}(\tau^{\prime})$ is an isomorphism onto $p_{\mathfrak{X}}^{-1}(\tau)$ we have

$$c_{1}\left(\overline{\mathcal{O}}^{h_{i}\circ p_{\mathfrak{X}}}\right)^{n}=\left(\varphi\Big|_{p_{\mathfrak{X}^{\prime}}^{-1}(\tau^{\prime})}\right)_{\ast}c_{1}\left(\left(\varphi\Big|_{p_{\mathfrak{X}^{\prime}}^{-1}(\tau^{\prime})}\right)^{\ast}\overline{\mathcal{O}}^{h_{i}\circ p_{\mathfrak{X}}}\right)^{n}$$

By Corollary 5.5 we have

$$c_{1}\left(\left(\varphi\Big|_{p_{\mathfrak{X}^{\prime}}^{-1}(\tau^{\prime})}\right)^{\ast}\overline{\mathcal{O}}^{h_{i}\circ p_{\mathfrak{X}}}\right)^{n}=c_{1}\left(\overline{\mathcal{O}}^{h_{i}\circ\varphi^{\textup{an}}\circ p_{\mathfrak{X}^{\prime}}}\right)^{n}=n!\cdot\MA\left(h_{i}\circ\varphi^{\textup{an}}\Big|_{\tau^{\prime}}\right).$$

Hence

$$c_{1}\left(\overline{\mathcal{O}}^{h_{i}\circ p_{\mathfrak{X}}}\right)^{n}=\left(\varphi\Big|_{p_{\mathfrak{X}^{\prime}}^{-1}(\tau^{\prime})}\right)_{\ast}\left(n!\cdot\MA\left(h_{i}\circ\varphi^{\textup{an}}\Big|_{\tau^{\prime}}\right)\right)=n!\cdot\MA(h_{i}).$$

It is easily seen that in Proposition 4.13 we can replace uniform convergence by locally uniform convergence. The claim follows from this fact and continuity of the real Monge-Ampère operator. ∎