The cluster-algebra tropical fan conjecture

Let $\mathcal{A}$ be a cluster algebra of finite type over $\mathbb{R}$, let $C$ be its coefficient-variable set, and let $S(\mathcal{A})$ be its associated cluster complex. Consider the positive tropical variety $\operatorname{Trop}^+ \operatorname{Spec} \mathcal{A}$ and its lineality space.

Tropical fan conjecture. If the lineality space of $\operatorname{Trop}^+ \operatorname{Spec} \mathcal{A}$ has dimension $|C|$, then the quotient of $\operatorname{Trop}^+ \operatorname{Spec} \mathcal{A}$ by its lineality space is a simplicial fan abstractly isomorphic to the cone over $S(\mathcal{A})$. If this dimension condition does not hold, the resulting fan is a coarsening of the cone over $S(\mathcal{A})$.

This conjecture proposes that using enough coefficient variables recovers the full cluster complex from the positive tropical variety, while insufficient coefficients produce a coarsening. The statement is motivated by the finite-type Grassmannian examples and by the observed failure for the coefficient-specialized cluster algebra of $Gr_{2,6}$; its general status is not established in the supplied source.