The generalized cluster complex shellability conjecture

Let $\Phi$ be a finite crystallographic root system of rank $n$, and let $m$ be a positive integer. The generalized cluster complex $\Delta^m(\Phi)$ is the simplicial complex associated with $\Phi$ and $m$. Shellability conjecture. The complex $\Delta^m(\Phi)$ is shellable, hence Cohen–Macaulay, and homotopy equivalent to a wedge of spheres. This was suggested by discussions with Hugh Thomas in December 2004. It has been proved by E. Tzanaki in types $A_n$ and $B_n$, while the statement remains open in the other types covered by the paper.