Hausel–Letellier–Rodriguez-Villegas conjecture for branched character varieties

Let $\mathcal{C}$ be a generic tuple of conjugacy classes, let $\bs\mu$ encode the eigenvalue multiplicities, and let $\mathcal{M}_{\mathcal{C}}$ be the associated character variety of dimension $d_{\bs\mu}$. Let $\mathbb{H}_{\bs\mu}(z,w)$ be the rational function defined from the Macdonald-polynomial generating series, and write $q=xy$. Hausel–Letellier–Rodriguez-Villegas conjecture. The following hold: $\mathbb{H}_{\bs\mu}(z,w)$ is a polynomial of degree $d_{\bs\mu}$ in each variable, with $\mathbb{H}_{\bs\mu}(-z,w)$ having nonnegative integer coefficients; $H_c(\mathcal{M}_{\mathcal{C}};x,y,t)$ is a polynomial in $q$ and $t$; and

$$H_c(\mathcal{M}_{\mathcal{C}};q,t)=(t\sqrt{q})^{d_{\bs\mu}}\mathbb{H}_{\bs\mu}(-t\sqrt{q},1/\sqrt{q}).$$

In particular, the mixed Hodge polynomial depends only on $\bs\mu$, not on the generic eigenvalues. This is the conjectural mixed-Hodge-polynomial formula arising from point counting for character varieties; part (ii) has been proved by Mellit, while the full package stated here is not established in the supplied text.