Baraglia–Schaposnik's Nadler-group conjecture for Langlands dual branes

Let $G_c$ be a complex reductive group with Langlands dual group $^LG_c$. Let $i_1$ be the involution whose fixed points form a $(B,A,A)$-brane in $\mathcal{M}_{G_c}$, and let $\check{H}$ be the Nadler group associated with the Lie algebra $\check{\mathfrak{h}}$. Baraglia–Schaposnik's conjecture. The support of the dual brane to the fixed point set of $i_1$ is the moduli space

$$\mathcal{M}_{\check{H}}\subset \mathcal{M}_{^LG_{c}}$$

of $\check{H}$-Higgs bundles. This proposes that Langlands duality identifies the dual of the brane from $i_1$ with a moduli space governed by the Nadler group. Its resolution is not specified in the source.