If there exists such a SCR polyhedral complex $\Pi$, then $\Pi$ and $\psi$ determine a model of $\mathcal{O}(D_{\Psi})$ and hence a toric algebraic metric $\|\cdot\|$. By Proposition 5.41, $\psi=\psi_{\|\cdot\|}$ and, by the classification of toric metrics in Proposition 5.16, the function $\psi_{\|\cdot\|}-\Psi$ extends to a continuous function on $N_{\Sigma}$. ∎