Simple-polytope conjecture for linear LP algebras

Let ${\mathcal A}$ be a linear LP algebra of rank $r$. Simple-polytope conjecture. There exists a convex simple polytope $P({\mathcal A})$ of dimension $r$ such that its vertices are in bijection with the clusters of ${\mathcal A}$ and its facets are in bijection with the cluster variables of ${\mathcal A}$. Under these bijections, for every face $F$ of $P({\mathcal A})$, the facets containing $F$ are exactly the cluster variables contained in the intersection of the clusters corresponding to the vertices of $F$.

This conjecture would realize the cluster complex of every linear LP algebra by a convex simple polytope, strengthening the preceding graph-LP-algebra claim. The source gives no resolution.