Let $G$ be a complex reductive algebraic group and $X$ be a smooth compact complex curve. It's easy to see that the space of vacua in B-twisted $N=4$ SUSY Yang--Mills theory is $\mathfrak{h}^*[2]/W$ (where $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{g}$ and $W$ is the Weyl group).

There is a theorem due to Elliott and Yoo stating that the full category of boundary conditions compatible with the vacuum $0 \in \mathfrak{h}^*[2]/W$ is equivalent to $IndCoh_{\mathcal{N}_G}(LocSys_G)$, the category of ind-coherent sheaves on the moduli space of principal $G$-bundles equipped with flat connection on $X$ with singular support contained in the global nilpotent cone. Note that the former is the spectral side of geometric Langlands correspondence due to Arinkin--Gaitsgory.

This theorem gives a physical interpretation to $IndCoh_{\mathcal{N}_G}(LocSys_G)$ (and makes it clear that it's the right category to consider in geometric Langlands correspondence, as opposed to the category of quasi-coherent sheaves $QCoh(LocSys_G)$). Elliott and Yoo formulated following conjecture expressing the effect of gauge symmetry breaking on the categories under consideration: the full subcategory of objects in $IndCoh(LocSys_G)$ compatible with a vacuum $u\in \mathfrak{h}^*/W$ is equivalent to $IndCoh_{\mathcal{N}_L}(LocSys_L)$ where $L$ is the stabilizer of $u$.

My question is: what are the mathematical implications of this conjecture in the context of geometric Langlands program?