The positive-region coheight formula for q,t-Fuss-Catalan numbers

Let $W$ be a crystallographic reflection group, let $\mathsf{Shi}^{(m)}(W)$ be the extended Shi arrangement, and let $\operatorname{coh}(R)$ be the coheight of a positive region $R$. Positive-region formula. The $q,t$-Fuss-Catalan numbers specialize at $t=1$ to

$$\operatorname{Cat}^{(m)}(W;q,1)=\operatorname{Cat}^{(m)}(W;q):=\sum_R q^{\operatorname{coh}(R)},$$

where the sum is over all positive regions of $\mathsf{Shi}^{(m)}(W)$. This gives a conjectural combinatorial interpretation in terms of positive Shi regions; the source gives no resolution.