Gaitsgory's factorization conjecture for local systems on the formal punctured disc

Let $G$ be a reductive group and let $D^{\circ}$ be the formal punctured disc. Write $\operatorname{LocSys}_G(D^{\circ})$ for the stack of $G$-local systems on $D^{\circ}$, and let $\operatorname{Rep}(G)$ denote the symmetric monoidal DG category of representations of $G$. Let $\mathbf{ModCat}$ be the $(\infty,2)$-category of module categories and let $\mathbf{FactModCat}$ denote the corresponding category of factorization module categories.

Gaitsgory's factorization conjecture. There exists a fully faithful functor

$$\bFact: \operatorname{QCoh}(\operatorname{LocSys}_G(D^{\circ}))\mathbf{-ModCat} \longrightarrow \operatorname{Rep}(G)\mathbf{-FactModCat}$$

that is compatible with the forgetful functors from both sides to $\mathbf{DGCat}$. On the left, $\operatorname{QCoh}(\operatorname{LocSys}_G(D^{\circ}))$ is viewed as a monoidal category with its usual tensor product, and the left-hand side is the $(\infty,2)$-category of modules for this monoidal category. On the right, $\operatorname{Rep}(G)$ is equipped with its factorization-algebra category structure induced by its symmetric monoidal structure.

This conjecture gives a factorization-theoretic categorical incarnation of the local geometric Langlands perspective and relates categorical representations to local systems with restricted variation. The paper presents it as a conjectural structural statement; its resolution status is not specified in the supplied text.