Perverse topological mirror symmetry for rank two character varieties

Let $C$ be a compact Riemann surface of genus $g$, let $r$ be the rank, and let $M_{\mathrm{Dol}}(C,\operatorname{SL}_r)$ and $M_{\mathrm{Dol}}(C,\operatorname{PGL}_r)$ be the corresponding Dolbeault moduli spaces. Set

$$\dim=2(r^2-1)(g-1).$$

Let $PIE$ be the perverse intersection E-polynomial, let $PIE_{\mathrm{st}}$ be the stringy perverse intersection E-polynomial, and let $\Gamma=\operatorname{Pic}^0(C)[r]$ act on $M_{\mathrm{Dol}}(C,\operatorname{SL}_r)$ by tensorisation.

Perverse topological mirror symmetry.

$$PIE(M_{\mathrm{Dol}}(C,\operatorname{SL}_r);u,v,q)=(uvq)^{\dim}PIE_{\mathrm{st}}\left(M_{\mathrm{Dol}}(C,\operatorname{PGL}_r);u,v,\frac{1}{uvq}\right).$$

The conjecture predicts an exchange of the perverse Hodge numbers under topological mirror symmetry. Its specialization at $q=1$ recovers the degree-zero intersection-E-polynomial identity, while relative hard Lefschetz supplies the corresponding duality for the perverse variable.