Hausel–Rodriguez-Villegas mixed Hodge polynomial conjecture

Let $\mathcal{M}_n$ be the moduli space under consideration, let $d_n$ denote its dimension, and let $\mathcal{H}_{\lambda}(z,w)$, $\overline{H}_n(z,w)$, and $H(\mathcal{M}_n;q,t)$ be the genus-$g$ hook function, the associated rational function, and the mixed Hodge polynomial, respectively. Hausel–Rodriguez-Villegas conjecture. The mixed Hodge polynomial of $\mathcal{M}_n$ is given by

$$H(\mathcal{M}_n;q,t)=\big(t q^{1/2}\big)^{d_n}, \overline{H}_nbig( q^{1/2},-t^{-1} q^{-1/2}\big).$$

The conjecture is known in genus $0$, where it reduces to a Macdonald-polynomial identity, and the paper discusses a non-rigorous string-theoretic derivation in more general settings. Its general validity is presented as an open conjecture.