Geometric Langlands connected-components conjecture

Let $X$ be a smooth complete curve over a field $k$, let $G$ be a reductive group, and let $\Shv_{\Nilp}(\Bun_G)$ be the category of $\ell$-adic sheaves on $\Bun_G$ with nilpotent singular support. Let $\LS^{\on{restr}}_\cG$ be the prestack of $\cG$-local systems on $X$ with restricted variation, and let $'\!\LS^{\on{restr}}_\cG$ be the union of some of its connected components. Geometric Langlands connected-components conjecture. The inclusion

$$'\!\LS^{\on{restr}}_\cG\subset \LS^{\on{restr}}_\cG$$

is always an equality. The main theorem establishes this equality when $\operatorname{char}(k)=0$ and, for arbitrary $k$, when $G=GL_n$; the conjecture asks for the equality in all characteristics and for all relevant groups.