Etingof's monodromy conjecture for Dunkl-operator eigenlines

Let $W$ be a Weyl group, let $C_\Delta$ be the cactus group acting on $W$, and suppose that the family of algebras $D_\chi$, parametrized by $\chi\in\mathfrak h^{reg}$, compactifies to a family parametrized by $\mathcal M_\Delta$. Assuming a simple-spectrum result, this gives a covering of $\mathcal M_\Delta(\mathbb R)$ whose fibres are the eigenlines $\mathcal E_{D_\chi}(\mathbb C W)$. Etingof's monodromy conjecture. The monodromy of this covering agrees with the action of $C_\Delta$ on $W$. This conjecture relates the monodromy of the eigenline covering to the cactus-group action constructed using perverse equivalences and wall-crossing functors; the source does not state whether it has been proved or disproved.