Mixed-Hodge-polynomial conjecture for wreath character varieties

Let $\mathcal{C}=(C_j)_{1\le j\le 2k}$ be a strongly generic tuple of semisimple conjugacy classes in $\operatorname{GL}_n\langle\sigma\rangle$ satisfying the condition referred to as $\mathrm{CCL}$. Let $\mathbf{B}$ be the tuple of types associated to the classes, let $\operatorname{Ch}_{\mathcal{C}}$ be the associated character variety of dimension $d$, and let $\mathbb{H}_{\mathbf{B}}(z,w)$ be the rational function defined in the source. Write $q=xy$. Wreath-character-variety mixed-Hodge-polynomial conjecture. The function $\mathbb{H}_{\mathbf{B}}(z,w)$ is a polynomial of degree $d$ in each variable, every monomial has even degree, and $\mathbb{H}_{\mathbf{B}}(-z,w)$ has nonnegative integer coefficients; $H_c(\operatorname{Ch}_{\mathcal{C}};x,y,t)$ is a polynomial in $q$ and $t$; and

$$H_c(\operatorname{Ch}_{\mathcal{C}};q,t)=(t\sqrt{q})^d\mathbb{H}_{\mathbf{B}}(-t\sqrt{q},1/\sqrt{q}).$$

In particular, it depends only on $\mathbf{B}$ and not on the generic eigenvalues. Part (i) gives a sufficient condition for the displayed expression to have nonnegative integer coefficients, while part (ii) says the mixed Hodge structure is Hodge–Tate; no resolution is given in the supplied text.