Crystal conjecture for toric non-archimedean μ-entropy maximizers

Let $\mathcal{P}$ be a system, let $\lambda\in\mathbb{R}$, and let $q$ be a maximizer of the toric non-archimedean $\mu$-entropy functional $\bm{\check{\mu}}_{\mathrm{NA}}^\lambda$. Crystal conjecture. For every $\lambda\in\mathbb{R}$, every such maximizer $q$ is piecewise affine. This conjecture predicts that maximizers are realized by piecewise-affine convex functions, which correspond in the toric setting to algebro-geometric degenerations; the source relates the expected regularity to moduli theory for polarized varieties and notes that the conjecture is motivated by analogous results in the Kähler–Ricci soliton context.