Let $X$ be an $n$-dimensional proper variety over $K$ and $\overline{L}$ a line bundle with a fixed formal metric. Let $\mu$ be a positive Borel measure on $X^{\textup{an}}$ and $\varphi$ a continuous function on $X^{\textup{an}}$ such that the metric on $\overline{L}\otimes\overline{\mathcal{O}}^{\varphi}$ is semipositive and solving the equation

$$c_{1}(\overline{L}\otimes\overline{\mathcal{O}}^{\varphi})^{n}=\mu.$$

Let $\tau$ be an $n$-dimensional open face of some skeleton $\Delta$ associated to a strongly nondegenerate strictly polystable formal model $\mathfrak{X}$ of $X^{\textup{an}}$. Suppose that $\mathfrak{X}$ is algebraic, $\overline{L}$ has a model on $\mathfrak{X}$ and $\lambda\cdot\boldsymbol{dx}\leq\mu\leq\Lambda\cdot\boldsymbol{dx}$ on $\tau$ for some $\lambda,\Lambda>0$ where $\boldsymbol{dx}$ denotes the Lebesgue measure on $\tau$. Assume that $\varphi=\varphi\circ p_{\mathfrak{X}}$. Then $\varphi\in W^{2,1}_{loc}(\tau)$.