Crystallographic q-specialization conjecture for q,t-Fuss-Catalan numbers

Let $W$ be a crystallographic reflection group, and let $\operatorname{Cat}^{(m)}(W,q)$ be the combinatorial $q$-Fuss-Catalan number obtained by summing $q^{\operatorname{coh}(R)}$ over the positive regions $R$ of the extended Shi arrangement. Crystallographic q-specialization conjecture.

$$\operatorname{Cat}^{(m)}(W,q,1)=\operatorname{Cat}^{(m)}(W,q).$$

The conjecture would partially answer the paper's open problem concerning combinatorial statistics realizing the two-parameter polynomial. The type $A$ result for all $m$ and the type $B$ result for $m=1$ are established in the supplied context, but the stated general crystallographic claim remains unresolved there.