Dolbeault topological mirror symmetry conjecture for character varieties

Let $n$ be a positive integer, let $d,e$ be integers satisfying $(d,n)=(e,n)=1$, and let $\mathcal{M}_{\Dol}^d$ and $\hat{\mathcal{M}}_{\Dol}^e$ denote the corresponding Dolbeault moduli spaces for $\operatorname{GL}_n$ and $\operatorname{PGL}_n$. Let $\Gamma=\mathbb{Z}_n^{2g}$ act on the relevant moduli space, and let $\hat{B}^d$ be the associated $\Gamma$-equivariant gerbe on $\hat{\mathcal{M}}_{\Dol}^e$. The stringy $E$-polynomial with this gerbe twist is denoted by $E_{\st}^{\hat{B}^d}$. Dolbeault topological mirror symmetry conjecture. For $(d,n)=(e,n)=1$, one has

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

This is the Dolbeault form of the proposed topological mirror symmetry between the $\operatorname{GL}_n$ and $\operatorname{PGL}_n$ Hitchin systems. The source presents it as equivalent to the de Rham version using the equality of the corresponding mixed Hodge structures; its resolution is not specified here.