Corti's cluster-variety conjecture for toric specializations of Fano varieties

Let $X$ be a generic Fano variety with klt singularities. A polarized cluster variety is a cluster variety equipped with a polarization; write it as $U=Y\setminus D$ with polarization $p$, and let $\mathbb{T}^\star$ denote the source of a torus chart. For each torus chart $j\colon \mathbb{T}^\star \dasharrow U$, let $X_{p(j)}$ be the toric Fano variety associated with the Fano polytope $p(j)$. A toric specialization of $X$ means a toric degeneration of $X$. Corti's conjecture. There is a polarized cluster variety $U=Y\setminus D$ with polarization $p$ such that the set

$$\bigl\{X_{p(j)} \mid \text{$j\colon \mathbb{T}^\star \dasharrow U$ is a torus chart}\bigr\}$$

up to isomorphism is precisely the set of toric specializations of $X$ up to isomorphism. This conjecture organizes toric degenerations of generic klt Fano varieties through cluster structures; the paper states that it is proved in dimension $2$, while higher-dimensional cases are supported by evidence from the Fanosearch program, cluster structures, and moduli spaces of conformal blocks.