Kapustin–Witten Langlands duality conjecture for cohomological Donaldson–Thomas invariants

Let $G$ be a reductive group, $G^{L}$ its Langlands dual, and $M$ a closed oriented $3$-manifold. Kapustin–Witten Langlands duality conjecture. There is an isomorphism

$$\HHf(\Loc_{G}(M),\varphi_{G}(M)) \cong \HHf(\Loc_{G^{L}}(M), \varphi_{G^{L}}(M)).$$

This is motivated by S-duality for the Kapustin–Witten theories, which exchanges a reductive group with its Langlands dual. The conjecture predicts a corresponding duality for the cohomological Donaldson–Thomas theories of local systems on closed oriented $3$-manifolds.