Langlands duality for twisted stringy BPS cohomology of Higgs moduli

Let $C$ be a smooth projective curve and let $G$ be a semisimple group, with Langlands dual group $G^{\vee}$. Let $\mathrm{H}^*_{\mathrm{BPS},\mathrm{st},\mathrm{tw}}(\mathrm{H}^{\mathrm{ss}}_G)$ denote the twisted stringy BPS cohomology of the semistable $G$-Higgs moduli space. Dolbeault Langlands duality conjecture. There exists a natural isomorphism

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

This is proposed as the Dolbeault counterpart of Langlands duality for twisted stringy BPS cohomology of character stacks, motivated by compatibility with non-abelian Hodge correspondence; its resolution status is not specified in the supplied text.