Langlands duality conjecture for character varieties of free groups

Let $\Gamma=F_r$ be the free group of rank $r$, and let $G$ and $G^L$ be complex reductive Langlands dual groups. Write $\mathcal{X}_rG$ and $\mathcal{X}_rG^L$ for their character varieties, and let $e(-)$ denote the $E$-polynomial. A variety is of Hodge–Tate type when its mixed Hodge structure has Hodge numbers vanishing off the diagonal. Langlands duality conjecture. Both character varieties are of Hodge–Tate type, and

$$e(\mathcal{X}_rG)=e(\mathcal{X}_rG^L).$$

The conjecture is motivated by the equality proved in the paper for $G=SL_n$ and $G^L=PGL_n$, and it predicts a corresponding equality and Hodge–Tate property for all complex reductive Langlands dual groups. The general statement was left for future work.