Under the assumptions (A1)–(A3), let $\theta$ be a closed $(1,1)$-form on ${X^{{\mathrm{an}}}}$ with ample de Rham class $\{\theta\}$ and let $\mu$ be a positive Radon measure on $X^{\mathrm{an}}$ of mass $\{\theta\}^{n}$ supported on the skeleton of a projective SNC-model. Then there is a continuous $\theta$-psh function $\varphi$ on ${X^{{\mathrm{an}}}}$ such that ${\rm MA}_{\theta}(\varphi)=\mu$ and $\varphi$ is unique up to additive constants.