(Sketch) For any fixed $x\in N_{\mathbb{R}}$, the function $\langle x,m\rangle+\lambda(m)$ is a concave function of $m\in M_{\mathbb{R}}$. By our assumptions, the set of $m\in\Delta_{\mathbb{Z}}$ saturating the maximum must be the set of vertices of some simplex $\sigma$ in the triangulation of $\Delta$. A more effective version of this observation is the Lemma in the $\mathcal{A}^{\infty}_{\lambda}$ case, and the $\mathcal{A}^{s}_{\lambda}$ case follows by Prop. 3.2. ∎