The Gluing Conjecture for the global nilpotent cone

Let $G$ be a reductive group, let $X$ be a curve, and let $\operatorname{LocSys}_G$ be the algebraic stack of $G$-local systems on $X$. For each standard parabolic $P\subset G$, let $\operatorname{LocSys}_P$ be the stack of $P$-local systems, and write $\operatorname{Par}(G)$ for the poset of standard parabolics. Let $\operatorname{IndCoh}_{\operatorname{Nilp}_{\operatorname{glob}}}(\operatorname{LocSys}_G)$ denote the category of ind-coherent sheaves with singular support in the global nilpotent cone. The pullback and restriction functors define a functor

$$\operatorname{IndCoh}_{\operatorname{Nilp}_{\operatorname{glob}}}(\operatorname{LocSys}_G)\to \operatorname{Glue}(\operatorname{QCoh}(\operatorname{LocSys}_P)_{\operatorname{conn}/\operatorname{LocSys}_G},P\in \operatorname{Par}(G)).$$

Gluing Conjecture. This functor is fully faithful.

The conjecture asserts that objects with global nilpotent singular support can be recovered from their restrictions to the stacks of local systems for all standard parabolics, together with the corresponding homotopy-coherent compatibility data. The paper states that its goal is to prove this conjecture.