Speyer–Williams cluster-complex conjecture for finite-type cluster algebras

Let $\mathcal A$ be a cluster algebra of finite type, let $\mathcal S(\mathcal A)$ be its associated cluster complex, and let $C$ denote the relevant cluster-variable set whose cardinality is $|C|$. Consider the positive tropicalization $\operatorname{Trop}^+\operatorname{Spec}\mathcal A$. Speyer–Williams conjecture. If the lineality space of $\operatorname{Trop}^+\operatorname{Spec}\mathcal A$ has dimension $|C|$, then $\operatorname{Trop}^+\operatorname{Spec}\mathcal A$ is abstractly isomorphic to the cone over $\mathcal S(\mathcal A)$. If this lineality-space condition does not hold, the resulting fan is a coarsening of the cone over $\mathcal S(\mathcal A)$. This conjecture relates positive tropicalizations to cluster complexes and is known in the cases covered by Brodsky and Stump, including the Grassmannian $\operatorname{Gr}_0(3,8)$; the unrestricted finite-type formulation is resolved.