Dimension conjecture for q,t-Fuss-Catalan numbers of complex reflection groups

Let $W$ be a well-generated complex reflection group. The $q,t$-Fuss-Catalan number is defined by

$$\operatorname{Cat}^{(m)}(W,q,t)=\mathcal{H}(M^{(m)};q,t),$$

where $M^{(m)}$ is the associated $W$-module, so that $\operatorname{Cat}^{(m)}(W,1,1)=\dim M^{(m)}$, and

$$\operatorname{Cat}^{(m)}(W)=\prod_{i=1}^l\frac{d_i+mh}{d_i}$$

for rank $l$, fundamental-invariant degrees $d_1\leq\cdots\leq d_l$, and Coxeter number $h=d_l$. Dimension conjecture. For every well-generated complex reflection group $W$,

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

This conjecture is based on computer calculations for several classical and exceptional reflection groups and predicts that the specialization of the bigraded Hilbert series gives the Fuss-Catalan product.