Polytopality conjecture for multi-cluster complexes

Let $\Delta_c^k(W)$ be a multi-cluster complex of a finite Coxeter group $W$.

Multi-cluster polytopality conjecture. The multi-cluster complex $\Delta_c^k(W)$ is the boundary complex of a simplicial polytope.

The source presents this conjecture in the context of the polytopality of spherical subword complexes and notes that the preceding conjectures hold for dihedral types; no general resolution is given.