Speyer–Williams cluster-complex conjecture for positive tropicalizations

Let $\mathcal S_{k,n}$ denote the possibly infinite cluster complex of the Grassmannian $\operatorname{Gr}(k,n)$. Let $\operatorname{TGr}_0^+(k,n)$ denote its positive tropicalization, and consider all reembeddings of $\operatorname{TGr}_0(k,n)$ into $\mathbb R^{|\Lambda|}$, where $\Lambda$ is any finite set of cluster variables containing the Plücker variables. Speyer–Williams conjecture. The cluster complex is the inverse limit of these positive tropicalizations:

$$\mathcal S_{k,n} = \varprojlim \operatorname{TGr}_0^+(k,n).$$

The conjecture predicts that the combinatorial structure of the positive tropicalization is captured by the cluster complex, including for Grassmannians with infinite-type cluster algebras. It was proven by Brodsky and Stump for cluster algebras of type $A$ and for all types of rank at most $8$, including $\operatorname{Gr}_0(3,8)$; the general statement remains open.