The reciprocal-specialization formula for q,t-Fuss-Catalan numbers

Let $W$ be a well-generated complex reflection group of rank $\ell$, acting on $\mathbb{C}[V\oplus V^*]$. Let $d_i$ and $d_i^*$ be the invariant and codegrees appearing in the source, let $h$ be the Coxeter number, and set

$$N=\sum_i(d_i-1),\qquad N^*=\sum_i(d_i^*+1).$$

Here $N$ is the number of reflections and $N^*$ is the number of reflecting hyperplanes. Reciprocal-specialization conjecture.

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

This is intended to generalize the corresponding type $A$ formula and is stronger than the dimension conjecture; the source gives no resolution.