Betti topological mirror symmetry conjecture for character varieties

Let $\mathcal{M}_{\B}^d$ be the $\operatorname{GL}_n$ character variety with central monodromy $\zeta_n^d I_n$, and let $\hat{\mathcal{M}}_{\B}^e$ be the corresponding $\operatorname{PGL}_n$ character variety, obtained as the quotient by $\Gamma=\mathbb{Z}_n^{2g}$. Assume $(d,n)=(e,n)=1$, and let $\hat{B}^d$ be the $\Gamma$-equivariant gerbe on $\mathcal{M}_{\B}^e$ analogous to the gerbe introduced for the Higgs moduli spaces. Betti topological mirror symmetry conjecture. For $(d,n)=(e,n)=1$, one has

$$E(\mathcal{M}_{\B}^d;u,v)=E_{\st}^{\hat{B}^d}(\hat{\mathcal{M}}_{\B}^e;u,v).$$

This is the Betti, or character-variety, formulation of topological mirror symmetry. It follows formally from the corresponding de Rham formulation under the Riemann–Hilbert correspondence, while the supplied text does not state whether the conjecture has been resolved.