The Mackey formula for categories O of rational Cherednik algebras

Let $W$ be a finite complex reflection group, and let $W_a$ and $W_b$ be parabolic subgroups. Write ${}^{a}W^{b}$ for a complete set of double coset representatives of $W_a\backslash W/W_b$. The Bezrukavnikov–Etingof induction and restriction functors are denoted by $\,^{\mathcal{O}}\!\operatorname{Ind}$ and $\,^{\mathcal{O}}\!\operatorname{Res}$. The Mackey formula for $\mathcal{O}$. At the level of representation categories, there is an isomorphism of functors

$$\,^{\mathcal{O}}\!\operatorname{Res}^{W}_{W_a}\circ\,^{\mathcal{O}}\!\operatorname{Ind}^{W}_{W_b}\cong\bigoplus_{u\in{}^{a}W^{b}}\,^{\mathcal{O}}\!\operatorname{Ind}^{W_a}_{W_a\cap uW_bu^{-1}}\circ u(-)\circ\,^{\mathcal{O}}\!\operatorname{Res}^{W_b}_{u^{-1}W_au\cap W_b}.$$

This extends the classical Mackey formula to Bezrukavnikov–Etingof induction and restriction for categories $\mathcal{O}$ associated with rational Cherednik algebras; the paper reports proving it, but the supplied material does not establish whether the conjecture was subsequently resolved.