Bessis–Reiner cyclic sieving conjecture for nonnesting partitions

Let $W$ be a finite crystallographic Coxeter group with Coxeter number $h$, let $J(\Phi^+(W))$ be the order ideals of its positive root poset, and let $d_1,d_2,\ldots,d_n$ be the degrees of $W$. Define

$$Cat(W,q)=\prod_{i=1}^n\frac{[h+d_i]_q}{[d_i]_q}.$$

Let $C_{2h}$ be the cyclic group of order $2h$. Bessis–Reiner's conjecture. Let $C_{2h}$ act on $J(\Phi^+(W))$ by $\operatorname*{Row}$. Then $(J(\Phi^+(W)),Cat(W,q),C_{2h})$ exhibits the cyclic sieving phenomenon. The conjecture was proved by D. Armstrong, C. Stump, and H. Thomas through an equivariant bijection between nonnesting and noncrossing partitions.