Minimal non-face conjecture for multi-cluster complexes

Let $W$ be a finite Coxeter group, let $c$ be a Coxeter element, and let $\Delta_c^k(W)$ be the associated multi-cluster complex. A minimal non-face is an inclusion-minimal subset that is not a face.

Minimal non-face conjecture. Every minimal non-face of $\Delta_c^k(W)$ has cardinality $k+1$.

The lower bound $k+1$ is known in general, and the equality is established in types $A$ and $B$, for dihedral groups, and in the tested cases of ranks $3$ and $4$ with $k=2$; the general statement remains open.