Gauge symmetry breaking conjecture for vacuum-restricted geometric Langlands categories

Let $G$ be a reductive group, let $\mathfrak h$ be a Cartan subalgebra with Weyl group $W$, and let $v\in\mathfrak h^*/W$. Let $x\in\mathfrak g^*$ be a semisimple lift of $v$, and let $L$ be the centralizer of $x$ in $\mathfrak g^*$. Write $\operatorname{IndCoh}_{\operatorname{Arth}^v_G(C)(\!(t)\!)}(\operatorname{Flat}_G(C))(\!(t)\!)$ for the category obtained by restricting $\operatorname{IndCoh}(\operatorname{Flat}_G(C))(\!(t)\!)$ at the vacuum $v$, and let $\mathcal N_L$ denote the nilpotent singular-support condition for $L$. Gauge symmetry breaking conjecture. There is an equivalence

$$\operatorname{IndCoh}_{\operatorname{Arth}^v_G(C)(\!(t)\!)}(\operatorname{Flat}_G(C))(\!(t)\!)\simeq\operatorname{IndCoh}_{\mathcal N_L}(\operatorname{Flat}_L(C))(\!(t)\!).$$

This predicts that moving away from the zero vacuum replaces the geometric Langlands category for $G$ by the nilpotent singular-support category for the Levi subgroup determined by the semisimple vacuum. The paper presents this as a conjectural geometric description of the more general vacuum-compatible categories.