The pure-EKR conjecture for generalized cluster complexes

Let $\Phi$ be a finite root system of rank $n-1$, let $m$ be a positive integer, and let $\mathcal C(m,\Phi)$ denote the $m$-th generalized cluster complex, a flag pure simplicial complex of dimension $n-2$.

Generalized cluster complex conjecture. Every generalized cluster complex $\mathcal C(m,\Phi)$ is pure-EKR.

For type $A_{n-1}$, these complexes are the complexes of $(m+2)$-angulations of a convex $(mn+2)$-gon. The paper reports computational verification for $n+m\leq 8$ and all irreducible root systems of rank $n-1$, but the general assertion remains open.