Polarized relative Dolbeault geometric Langlands conjecture

Let $X=G/H$ be an affine homogeneous spherical variety with abelian regular centralizers and no type $N$ roots, and let $S_X$ be a polarizable symplectic dual representation of $G_X^\vee$. Choose a polarization

$$S_X=S_X^-\oplus S_X^+.$$

Let $p_*\mathcal O_{\mathcal P}$ be the period sheaf and let $\mathrm{DH}(G_X^\vee,\bigwedge^\bullet S_X^+)$ be the associated Dirac–Higgs bundle. Polarized relative Dolbeault geometric Langlands conjecture.

$$\operatorname{FM}_{\mathcal P_G}(p_*\mathcal O_{\mathcal P})\simeq\mathrm{DH}(G_X^\vee,\bigwedge^\bullet S_X^+).$$

The right-hand side is asserted to be independent of the choice of polarization. The conjecture is known in the paper's previously treated special cases but is not established in the stated generality.