The ULA module characterization conjecture for local systems

Let $\sG$ be an algebraic group, let $X$ be a smooth proper curve, and set $\CA=\Rep(\sG)$. Let $\LocSys_{\sG}(X)$ be the stack of de Rham $\sG$-local systems on $X$. The canonical functor

$$\on{Fact}(\Rep(\sG))_{\Ran(X)}\to \QCoh(\LocSys_{\sG}(X))$$

induces a fully faithful functor

$$\QCoh(\LocSys_{\sG}(X))\mmod^{\on{c.g.}}\to \CA^{\otimes X}\mmod^{\on{c.g.}}.$$

ULA module characterization conjecture. This functor induces an equivalence onto

$$\CA^{\otimes X}\mmod^{\on{ULA}}\subset \CA^{\otimes X}\mmod^{\on{c.g.}}.$$

The proposition immediately before the conjecture establishes one containment, so the conjecture asserts that every compactly generated ULA module arises from $\QCoh(\LocSys_{\sG}(X))$; the source gives no resolution.