Reiner–Stanton–White cyclic sieving conjecture for facets of generalized cluster complexes

Let $s$ be a positive integer, let $\Phi$ be a root system of rank $n$ with Coxeter number $h$ and exponents $e_1,\ldots,e_n$, and let $X$ be the set of facets of the generalized cluster complex $\Delta^s(\Phi)$. Define

$$\operatorname{Cat}^{(s)}(\Phi,q):=\prod_{i=1}^n\frac{[sh+e_i+1]_q}{[e_i+1]_q}.$$

Let $X(q)=\operatorname{Cat}^{(s)}(\Phi,q)$, and let the cyclic group $C$ of order $sh+2$, generated by $\Gamma_s$, act on $X$. Reiner–Stanton–White's cyclic sieving conjecture. The triple $(X,X(q),C)$ exhibits the cyclic sieving phenomenon.

This conjecture concerns the cyclic action on facets of generalized cluster complexes, with the generalized $q$-Catalan polynomial serving as the sieving polynomial. The paper states that it proves the conjecture for generalized cluster complexes in types $A_n$, $B_n$, $D_n$, and $I_2(a)$; the supplied span does not establish a resolution for all root-system types.