Parabolic Coxeter–Catalan numbers for types A, B, H and I

Let $(W,S)$ be a Coxeter system with $W\in\{A_n,B_n,H_3,I_2(m)\}$, and let $J\subseteq S$. The sets $\operatorname{Align}(W^J,c)$, $\operatorname{NC}(W^J,c)$, $\operatorname{SW}(W^J,c)$, and $\operatorname{NN}(W^J)$ denote, respectively, the parabolic aligned elements, noncrossing elements, cluster-complex facets, and nonnesting elements. Parabolic Coxeter–Catalan conjecture. For any Coxeter element $c\in W$, the cardinalities

$$\left|\operatorname{Align}(W^J,c)\right|=\left|\operatorname{NC}(W^J,c)\right|=\left|\operatorname{SW}(W^J,c)\right|=\left|\operatorname{NN}(W^J)\right|$$

are equal, and hence do not depend on the choice of $c$. The claim proposes well-defined parabolic Coxeter–Catalan numbers in these types; the paper reports computer evidence, while no general proof or resolution is given in the supplied text.