Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a connected smooth complete curve over $k$. Consider the moduli stack $\mathrm{Bun}_G$ of principal $G$-bundles on $X$ for connected reductive group $G$.

Geometric Langlands conjecture states (among other things) that the DG category $D(\mathrm{Bun}_G)$ of D-modules on $\mathrm{Bun}_G$ should be equivalent to the DG category of ind-coherent sheaves on the moduli stack of $\check{G}$-local systems with singular support contained in the global nilpotent cone. Our ability to work with $D(\mathrm{Bun}_G)$ in practice depends on it admitting a set of compact generators.

Drinfeld and Gaitsgory have shown that for $X$ and $G$ satisfying the conditions above, the category $D(\mathrm{Bun}_G)$ is compactly generated. This poses a natual question: is the category $D(\mathrm{Bun}_G)$ compactly generated for any connected affine algebraic group $G$? My question is: has there been any progress on this question since Drinfeld--Gaitsgory (or maybe people have constructed counterexamples)?


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