Refined topological mirror symmetry conjecture for Hitchin moduli spaces

Let $\Gamma=H^1(C,\mathbb{Z}_n)$ act on the relevant Hitchin moduli space $\mathcal{M}^d$, and let $\hat\Gamma$ be its character group. Write $E_\kappa(\mathcal{M}^d;u,v)$ for the contribution of the $\kappa$-isotypic part of compactly supported cohomology, let $\hat B^d$ be the stated $\Gamma$-equivariant gerbe, and let $w:\Gamma\rightarrow\hat\Gamma$ be the identification induced by the Weil pairing. Refined topological mirror symmetry conjecture. For $\kappa\in\hat\Gamma$ we have

$$E_{\kappa}(\mathcal{M}^d;u,v)=E(\mathcal{M}^e_{\gamma}/\Gamma,L_{\hat B^d,\gamma};u,v)(uv)^{F(\gamma)},$$

where $\gamma=w(\kappa)$. This refines the unrefined topological mirror symmetry equality by matching each character contribution with a corresponding twisted sector. The source states that it holds for $n=2,3$; the general statement remains open.