Let $X$ be a smooth projective $K$-variety. Let $\mathcal{X}$ be a SNC model of $X$ and $\theta\in\mathcal{Z}^{1,1}(X)$ a closed $(1,1)$-form determined on $\mathcal{X}$. Then there exists a constant $C=C(\mathcal{X},\theta)>0$ such that for every $\theta$-psh model function $\varphi$, the composition $\varphi\circ\emb_{\mathcal{X}}$ is convex, piecewise affine and $C$-Lipschitz continuous on each face of $\Delta_{\mathcal{X}}$.