Baraglia–Schaposnik's Langlands duality conjecture for the $(A,B,A)$-brane

Let $G_c$ be a complex reductive group with Langlands dual group $^LG_c$, let $f$ be the anti-holomorphic involution of the Riemann surface defining $i_2$, and let $^L\rho$ be the compact structure of $^LG_c$. The involution $i_2$ acts on the Higgs-bundle moduli space by combining pullback by $f$ with the compact structure and a sign change on the Higgs field. Baraglia–Schaposnik's conjecture. The support of the dual brane of the fixed point set of $i_2$ is the fixed point set in $\mathcal{M}_{^LG_c}$ of

$$^Li_2(\bar \partial_A, \Phi)= (f^*(^L\rho(\bar \partial_A)), -f^*( ^L\rho(\Phi) )).$$

This conjecture predicts how Langlands duality exchanges the brane associated with the involution $i_2$ and a brane in the moduli space for the Langlands dual group. Its resolution is not specified in the source.