Cluster complex face-lattice conjecture

Let $\Phi$ be a root system of rank $n$, and let $\Delta(\Phi)$ be its cluster complex. Regard $\Delta(\Phi)$ as a poset under reverse inclusion.

Cluster complex face-lattice conjecture. This poset is the face lattice of a simple $n$-dimensional convex polytope $P(\Phi)$.

This is presented as a weaker version of the cluster fan polytopality conjecture. It seeks a simple polytope whose face structure is encoded by the cluster complex; the source supplies no resolution status.