Cluster fan polytopality conjecture

Let $\Phi$ be a root system of rank $n$, let $Q_{\mathbb R}$ be the ambient real vector space of its root lattice, and let $\Delta(\Phi)$ be the cluster complex. The cones $\mathbb R_{\geq 0}C$ generated by clusters $C$ form the complete simplicial fan described in the source.

Cluster fan polytopality conjecture. The simplicial fan generated by the clusters is the normal fan of a simple $n$-dimensional convex polytope $P(\Phi)$.

This conjecture asks for a polytopal realization of the cluster fan and connects cluster complexes with the geometry of generalized associahedra. The source supplies no resolution status.