1-affineness conjecture for the local space of systems on the punctured disc

Let $\CK$ be the Laurent series field of the formal punctured disc, let $\oD$ denote the formal punctured disc, let $G$ be the reductive group with Lie algebra $\cg$, and define the prestack of local systems by

$$\LocSys_\cG(\oD)=\cg\otimes\omega_\CK/G(\CK).$$

A prestack is 1-affine when the functors between sheaves of categories and module categories over its quasi-coherent category are mutually inverse equivalences.

Local 1-affineness conjecture. The prestack $\LocSys_\cG(\oD)$ is 1-affine.

This would identify sheaves of categories on the local space of systems with the corresponding module categories and is intended to support the local geometric Langlands framework; no resolution is given here.