Topological mirror symmetry for quasi-BPS categories of Higgs bundles

Let $G$ be a reductive group, let $^{L}G$ be its Langlands dual, and let $(\chi,w)\in\pi_1(G)\times Z(G)^{\vee}$. Let $\mathbb{T}_{^{L}G}(w)_{-\chi}$ and $\mathbb{T}_G(\chi)_w$ denote the corresponding quasi-BPS categories, with nilpotent-support subcategories defined similarly. Topological mirror symmetry conjecture. There is an equivalence

$$\mathbb{T}_{^{L}G}(w)_{-\chi}\simeq\mathbb{T}_G(\chi)_w,$$

which restricts to an equivalence of the quasi-BPS categories with nilpotent singular supports. This is presented as a consequence of compatibility of the two semiorthogonal decompositions and generalizes the cited conjecture for Higgs-bundle mirror symmetry.