The q-Kreweras cyclic sieving conjecture for principal-in-a-Levi nilpotent orbits

Let $W$ be a Weyl group with Coxeter number $h$, let $m=sh+1$, and let $NC^{(s)}(W)$ denote the set of multichains $w_1\leq\cdots\leq w_s$ in the $s$-divisible noncrossing partition poset. For each $W$-orbit $[X]$ of subsets $X$ of a Cartan subalgebra, let $\mathfrak l_X$ be the corresponding Levi subalgebra and let $\0_X$ be the nilpotent orbit consisting of elements principal nilpotent in $\mathfrak l_X$. The cyclic group $\mathbb Z/sh\mathbb Z$ acts on $NC^{(s)}(W)$, and $\omega_d$ denotes a primitive $d$th root of unity. The q-Kreweras cyclic sieving conjecture. For each $W$-orbit $[X]$, $\mathrm{Krew}(\Phi,\0_X,m;q=\omega_d)$ counts the elements $w_1\leq\cdots\leq w_s\in NC^{(s)}(W)$ that are fixed by an element of order $d$ in the $\mathbb Z/sh\mathbb Z$-action and satisfy $V^{w_1}\in[X]$. This would provide the q-analogue of the Kreweras numbers and generalize the cyclic sieving result for Fuss–Catalan numbers.