By Corollary B.4 $\varphi$ is convex on every closed face of $\Delta$. Note that the metric on $\overline{L}$ is trivial on $p_{\mathfrak{X}}^{-1}(\tau)$. Hence we can apply Corollary 5.7 to get

$$\mu=c_{1}(\overline{L}\otimes\overline{\mathcal{O}}^{\varphi})^{n}=\Deg(S)\cdot n!\cdot\MA(\varphi)$$

on $\tau$ where $S$ is the stratum of $\tilde{\mathfrak{X}}$ corresponding to $\tau$. Now the claim follows from the corresponding fact in the real case [Moo15, Theorem 1.2]. ∎