Diagonal Catalan specialization conjecture for root posets

Let $\Phi$ be a root system, let $\mathcal{M}(\Phi)$ be its diagonal coinvariant ring, and define its bigraded Hilbert series by

$$\operatorname{Cat}(\Phi;q,t)=\mathcal{H}(\mathcal{M}(\Phi);q,t).$$

Let $\mathcal{I}(\Phi^+)$ denote the order ideals of the positive-root poset. Diagonal Catalan specialization conjecture. The specialization $t=1$ yields the size generating function of order ideals:

$$\operatorname{Cat}(\Phi^+;q,1)=\sum_{I\in\mathcal{I}(\Phi^+)}q^{|I|}.$$

This generalizes the result known for type $A_n$ and was conjectured for other reflection groups; the source does not report a resolution in general.