Langlands duality for twisted stringy BPS cohomology of character stacks

Let $\Sigma$ be a compact oriented $2$-manifold and let $G$ be a semisimple group, with Langlands dual group $G^{\vee}$. Let $\mathrm{H}^*_{\mathrm{BPS},\mathrm{st},\mathrm{tw}}(\mathrm{Loc}_G(\Sigma))$ denote the twisted stringy BPS cohomology defined using sufficiently divisible orbifold covers. Langlands duality conjecture. There exists a natural isomorphism

$$\mathrm{H}^*_{\mathrm{BPS},\mathrm{st},\mathrm{tw}}(\mathrm{Loc}_G(\Sigma))\cong\mathrm{H}^*_{\mathrm{BPS},\mathrm{st},\mathrm{tw}}(\mathrm{Loc}_{G^{\vee}}(\Sigma)).$$

The conjecture is motivated by the corresponding Langlands duality conjecture for BPS cohomology of $3$-manifolds; its resolution status is not specified in the supplied text.