The global-nilpotent gluing conjecture for local systems

Let $G$ be a reductive group and let $X$ be a curve. For each standard parabolic $P\subset G$, let $\operatorname{LocSys}_P$ and $\operatorname{LocSys}_G$ denote the corresponding stacks of local systems, and let $\operatorname{Nilp}_{\operatorname{glob}}\subset\operatorname{Sing}(\operatorname{LocSys}_G)$ be the global nilpotent cone. For every $P$, let $\{0\}\subset\operatorname{Sing}(\operatorname{LocSys}_P)$ be the zero section. The functor in question is

$$\operatorname{IndCoh}_{\operatorname{Nilp}_{\operatorname{glob}}}(\operatorname{LocSys}_G)\to \operatorname{Glue}(\operatorname{IndCoh}_{\{0\}}((\operatorname{LocSys}_P)_{\mathrm{dR}}\mathbin{\underset{(\operatorname{LocSys}_G)_{\mathrm{dR}}}{\times}}\operatorname{LocSys}_G),P\in\operatorname{Par}(G)^{\operatorname{op}}).$$

Global-nilpotent gluing conjecture. This functor is fully faithful.

The conjecture concerns recovery of the category with global nilpotent singular support from its parabolic local-system pieces. It was made by the authors and recorded as Conjecture 9.3.7 in the cited earlier work; its resolution is not given in the supplied text.