Reiner–Chapoton cyclic sieving conjecture for chains in dual Garside categories

Let $W$ be a well-generated finite complex reflection group with degrees $d_1,\dots,d_n$ and Coxeter number $h$, let $d$ be a regular degree, let $W'$ be the centralizer of a $d$-regular element, and set $d':=d/(d\wedge h)$. Let $N\geq 1$, let $Z'_{W'}(N)$ count complete weak $N$-chains of simples in the reduced dual Garside category associated with $W'$, and let $\mu_d$ act on the corresponding complete weak $Nd'$-chains in the original category. Reiner–Chapoton cyclic sieving conjecture. For an indeterminate $q$ and $[a]_q:=1+q+\dots+q^{a-1}$, the expression

$$\prod_{i=1}^n\frac{[d_i+(Nd'-1)h]_q}{[d_i]_q}$$

is a polynomial in $q$ whose value at $q=\zeta_d$ equals $Z'_{W'}(N)$. This predicts that the $\mu_d$-fixed-point counts realize a cyclic sieving phenomenon; the source presents it as joint work with Vic Reiner and gives no resolution.