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

Let $W$ be a well-generated complex reflection group of rank $l$, with fundamental-invariant degrees $d_1\leq\cdots\leq d_l$, Coxeter number $h=d_l$, and $N:=\sum_{i=1}^l(d_i-1)$. Define the $q$-integer by $[r]_q=(1-q^r)/(1-q)$. q-specialization conjecture.

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

This is stronger than the dimension conjecture and would generalize the paper's stated theorem, while providing a new answer to a question of Kriloff and Reiner. It is based on the same finite computations and remains open in the supplied source.