Compatibility of the Dolbeault Langlands equivalence with quasi-BPS decompositions

Let $G$ be a reductive group, let $^{L}G$ be its Langlands dual, and let $(\chi,w)\in\pi_1(G)\times Z(G)^{\vee}$. Consider the quasi-BPS semiorthogonal decompositions of the coherent category on semistable $^{L}G$-Higgs bundles and of the limit category for $G$-Higgs bundles. Compatibility conjecture. Under the Dolbeault geometric Langlands equivalence, the semiorthogonal decompositions on the two sides are compatible. This is expected as part of the categorical Langlands correspondence and would imply the corresponding equivalence of quasi-BPS categories.