Let $X$ be a smooth projective curve 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

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

If $\tau$ is an open face of the skeleton $\Delta$ of a strictly semistable algebraic model $\mathscr{X}$ of $X^{\textup{an}}$ on which $\overline{L}$ has an algebraic model, $\mu$ is supported on $\Delta$ and $\mu=f\cdot\boldsymbol{dx}$ on $\tau$ for some positive function $f\in C^{k}(\tau)$ where $\boldsymbol{dx}$ denotes the Lebesgue measure on $\tau$ then $\varphi\in C^{k+2}(\tau)$.