Facet-maximality conjecture for multi-cluster complexes

Let $(W,S)$ be a finite Coxeter system of rank $n$, let $N=\ell(w_\circ)$, and let $Q$ be a word in $S$ with $kn+N$ letters. Let $\Delta(Q,w_\circ)$ be the corresponding subword complex, and let $\Delta_c^k(W)$ be a multi-cluster complex.

Facet-maximality conjecture. The number of facets of $\Delta(Q,w_\circ)$ is at most the number of facets of $\Delta_c^k(W)$. Moreover, if the two numbers are equal, then $Q$ has the SIN-property.

The conjecture is known for dihedral types $I_2(m)$; the source explains this using cyclic polytopes and an upper-bound argument. It remains open in general.