Frenkel–Gaitsgory–Vilonen vanishing conjecture for Levi averaging

Let $G$ be the group in the paper, let $P$ be a standard proper parabolic with Levi subgroup $M$, and let $W$ be an irreducible local system on $X$ of rank $r=\dim V^\gamma$. For $\mu\in\pi_1^+(M)$, suppose its image in $\pi_1(G)$ is $d\theta$, where $c(P)$ is the constant associated with $P$. Let $\operatorname{Av}^{\mu}_W$ be the averaging functor defined in the paper. Frenkel–Gaitsgory–Vilonen vanishing conjecture. For every $\mu\in\pi_1^+(M)$ whose image in $\pi_1(G)$ equals $d\theta$ with $d>c(P)$, the functor $\operatorname{Av}^{\mu}_W$ vanishes identically. This generalizes the Vanishing Conjecture of Frenkel, Gaitsgory and Vilonen from the full group to standard proper Levi subgroups. The source gives no resolution, so the conjecture remains open.