The oper generation conjecture for irreducible local systems

Let $\operatorname{LocSys}_\cG^{\operatorname{irred}}$ be the open substack of irreducible $\cG$-local systems. For every finite set $I$ and map $\lambda^I:I\to\Lambda^+$, let $\operatorname{Op}(\cG)^{\operatorname{glob,irred}}_{\lambda^I}$ be the corresponding global oper space and let

$$\operatorname{sv}_{\lambda^I}:\operatorname{Op}(\cG)^{\operatorname{glob,irred}}_{\lambda^I}\to\operatorname{LocSys}_\cG^{\operatorname{irred}}$$

be the forgetful map. Oper generation conjecture. If $\mathcal F\in\operatorname{QCoh}(\operatorname{LocSys}_\cG^{\operatorname{irred}})$ satisfies $(\operatorname{sv}_{\lambda^I})^!(\mathcal F)=0$ for all finite sets $I$ and maps $\lambda^I:I\to\Lambda^+$, then $\mathcal F=0$. Equivalently, the essential images of the pushforwards $(\operatorname{sv}_{\lambda^I})_*$ generate $\operatorname{QCoh}(\operatorname{LocSys}_\cG^{\operatorname{irred}})$. The source states that this is a theorem for $G=GL_n$ and follows for classical groups from work cited there.