Topological mirror symmetry conjecture for Langlands dual Hitchin systems

Let $C$ be a smooth projective curve, let ${\mathcal M}$ be the smooth quasi-projective $\SL_n$ Hitchin moduli space, and let ${\mathcal M}^\vee={\mathcal M}/\Gamma$ be the quasi-projective orbifold for $\PGL_n$, where $\Gamma=Jac_C[n]$. Let $\alpha\in H^2({\mathcal M}^\vee,{\rm U}(1))$ be the gerbe determined by the universal bundle, and let $h^{p,q}({\mathcal M})$ and $h_{st,\alpha}^{p,q}({\mathcal M}^\vee)$ denote the mixed and twisted stringy Hodge numbers defined in the paper. Topological mirror symmetry conjecture. There is an agreement of Hodge numbers

$$h^{p,q}({\mathcal M})=h_{st,\alpha}^{p,q}({\mathcal M}^\vee).$$

This is a mathematically testable form of mirror symmetry for the proposed Langlands-dual Hitchin-system mirrors. The equality was proved for the corresponding Hodge numbers by Hausel and Thaddeus, so the conjectural statement is supported by a known resolution of the cited topological mirror-symmetry claim.