The averaging-functor vanishing conjecture

Let $X$ be a smooth projective curve of genus $g$, let $E$ be a local system on $X$, and let $\Bun_n$ be the moduli stack of rank $n$ vector bundles. For $d\in\mathbb Z^+$, let $\Mod_n^d$ parametrize triples $(\M,\M',\beta)$ as in the paper, and let $\on{Av}^d_E:\on{D}(\Bun_n)\to\on{D}(\Bun_n)$ be the averaging functor defined by

$$\F\longmapsto \hl_!\left(\hr{}^*(\F)\otimes \s^*({\mathcal L}^d_E)\right)[nd].$$

The averaging-functor vanishing conjecture. Assume that $E$ is irreducible and has rank greater than $n$. If

$$d>(2g-2)\cdot n\cdot \operatorname{rk}(E),$$

then the functor $\on{Av}^d_E$ is identically equal to zero. This is the explicit form of the vanishing conjecture proposed in the cited earlier work, and it supplies the vanishing input for the geometric Langlands argument; its resolution status is not specified in the supplied text.