The Betti Langlands duality conjecture for BPS cohomology

Let $M$ be a three-manifold and let $G$ and $G^{\vee}$ be Langlands dual groups. Write $\mathfrak{M}_G(M)$ and $\mathfrak{M}_{G^{\vee}}(M)$ for the corresponding stacks of local systems, and $\operatorname{\mathtt{BPS}}_{\mathfrak{M}_G(M)}$ and $\operatorname{\mathtt{BPS}}_{\mathfrak{M}_{G^{\vee}}(M)}$ for their BPS cohomology. Betti Langlands duality conjecture. There is an isomorphism

$$\operatorname{\mathtt{BPS}}_{\mathfrak{M}_G(M)}\cong\operatorname{\mathtt{BPS}}_{\mathfrak{M}_{G^{\vee}}(M)}.$$

This conjecture proposes a BPS-cohomological form of Betti Langlands duality for stacks of local systems on three-manifolds.