The determinantal-component dimension conjecture for generalized diagonal coinvariants

Let $W$ be a well-generated complex reflection group, let $D\!R^{(m)}(W)$ be the space of generalized diagonal coinvariants, and let $\mathbf{e}_{\epsilon}$ denote the idempotent projecting to the determinantal representation component. Let $\operatorname{Cat}^{(m)}(W)$ be the Fuss-Catalan number of $W$. Dimension conjecture.

$$\dim\mathbf{e}_{\epsilon}\bigl(D\!R^{(m)}(W)\bigr)=\operatorname{Cat}^{(m)}(W).$$

This conjecture motivates the generalized $q,t$-Fuss-Catalan numbers; the source gives no resolution.