Bessis's cyclic sieving conjecture for the modified action on $m$-divisible noncrossing partitions

Let $W$ be a well-generated complex reflection group with Coxeter number $h$, let $m\geq 0$, and let $NC^m(W)$ be the set of $m$-divisible noncrossing partitions. Define

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

Let $C'=\mathbb Z/(m+1)h\mathbb Z$ act on $NC^m(W)$ by the modified cyclic action described in the source.

Bessis's modified-action CSP conjecture. The triple

$$(X,X(q),C')=\left(NC^m(W),\mathrm{Cat}^m(W,q),\mathbb Z/(m+1)h\mathbb Z\right)$$

exhibits the cyclic sieving phenomenon.

The modified action is related to the action for $NC^{m+1}(W)$ and has an interpretation via the Garside automorphism of the divided category associated with the dual braid monoid. The source gives no resolution status.