Hausel–Letellier–Villegas' mixed Hodge polynomial formula for character varieties

Let ${\bf C}$ be a generic tuple of conjugacy classes of $\operatorname{GL}_n(\mathbb{C})$, with type ${\bm\omega}\in\overline{{\mathbf{T}}}^{o}$, and let $d_{\bf C}$ denote the dimension parameter of the associated character variety. Write $IH_c({\mathcal{M}}_{\overline{{\bf C}}};q,t):=IH_c({\mathcal{M}}_{\overline{{\bf C}}};\sqrt q,\sqrt q,t)$. Hausel–Letellier–Villegas' mixed Hodge polynomial conjecture. The mixed Hodge polynomial depends only on $q$ and $t$ and satisfies

$$IH_c({\mathcal{M}}_{\overline{{\bf C}}};q,t)=(t\sqrt q)^{d_{\bf C}}\mathbb{H}_{\bm\omega}\left(-\frac{1}{\sqrt q},t\sqrt q\right).$$

The formula is stated for generic tuples and is presented as the main conjecture; the source gives no resolution status.