Point-independence conjecture for tempered D-modules on Bun_G

Let $X$ be a smooth complete curve, let $G$ be a reductive group, and let $x\in X$. The category $\Dmod(\Bun_G)$ is tensored over $\QCoh(\cg^*/(\cG\times \BG_m))$, giving each object a support $\on{supp}^{x}(\CM)$. Define

$$\Dmod^x_{\on{temp}}(\Bun_G):=\\{\CM\in\Dmod(\Bun_G):\on{supp}^x(\CM)=\\{0\\}\\}.$$

Point-independence conjecture. The subcategory $\Dmod^x_{\on{temp}}(\Bun_G)\subset\Dmod(\Bun_G)$ is independent of the choice of the point $x\in X$. This is deduced in the source as a consequence of the geometric Langlands conjecture and its compatibility with geometric Satake, so it remains open there.