Safronov's Langlands duality conjecture for critical cohomology of character stacks

Let $M$ be a compact oriented $3$-manifold, let $G$ be a connected reductive algebraic group, and let $G^{\vee}$ be the Langlands dual of $G$. Write $\mathcal{L}\mathrm{oc}_G(M)$ for the moduli stack of $G$-local systems on $M$, and let $\mathrm{H}^*_{\mathrm{crit}}(\mathcal{L}\mathrm{oc}_G(M))$ denote its critical cohomology.

Safronov's conjecture. There is a natural isomorphism

$$\mathrm{H}^*_{\mathrm{crit}}(\mathcal{L}\mathrm{oc}_G(M)) \cong \mathrm{H}^*_{\mathrm{crit}}(\mathcal{L}\mathrm{oc}_{G^{\vee}}(M)).$$

This proposes a Langlands duality for the critical cohomology of character stacks of $3$-manifolds. The source presents it as an application of the BPS decomposition theorem; no resolution is specified here.