Microlocal characterization conjecture for Lusztig sheaves

Let $Q$ be a quiver, let $\boldsymbol{d}\in\mathbf{N}^I$ be a dimension vector, and let $\flat\in\{\operatorname{nil},\emptyset,(\operatorname{nil},1),1\}$. Let $E_{\boldsymbol{d}}$ be the representation space, $G_{\boldsymbol{d}}$ the base-change group, and $\Lambda_{\boldsymbol{d}}^{\flat}$ the corresponding conical Lagrangian variety. The category $\mathcal{P}_{\boldsymbol{d}}^{\flat}$ consists of Lusztig perverse sheaves.

Microlocal characterization conjecture. Any irreducible $G_{\boldsymbol{d}}$-equivariant perverse sheaf on $E_{\boldsymbol{d}}$ whose singular support is contained in $\Lambda_{\boldsymbol{d}}^{\flat}$ is in the category $\mathcal{P}_{\boldsymbol{d}}^{\flat}$.

This is intended to characterize Lusztig sheaves microlocally through their singular support. The supplied excerpt does not state a resolution of this general claim; earlier results in the paper establish special cases, including the finite-type and affine loop-free situation and the cases covered by the preceding proposition.