Hausel–Letellier–Villegas conjecture for character varieties

Let ${\bm \mu}=(\mu^1,\ldots,\mu^k)\in{\mathcal{P}_n}^k$ be a tuple of partitions, let ${\mathcal{M}}_{\bm \mu}$ be the corresponding $\operatorname{GL}_n(\mathbb{C})$ character variety of a genus-$g$ Riemann surface with $k$ punctures, and let

$$d_{\bm \mu}:=n^2(2g-2+k)-\sum_{i,j}(\mu_j^i)^2+2.$$

Write $H_c({\mathcal{M}}_{\bm \mu};x,y,t)$ for its compactly supported mixed Hodge polynomial and define $H_c({\mathcal{M}}_{\bm \mu};q,t):=H_c({\mathcal{M}}_{\bm \mu};\sqrt q,\sqrt q,t)$. Hausel–Letellier–Villegas conjecture. The polynomial $H_c({\mathcal{M}}_{\bm \mu};x,y,t)$ depends only on $xy$ and $t$, and

$$H_c({\mathcal{M}}_{\bm \mu};q,t)=(t\sqrt q)^{d_{\bm \mu}}\;\mathbb{H}_{\bm \mu}\left(-{\frac 1{\sqrt q}},t\sqrt q\right).$$

This conjecture generalizes the corresponding conjecture for the varieties ${\mathcal{M}}_n$ and passes several consistency checks, including the cases $k=1$, $\mu^1=(n)$ and $g=0$, $k=2$. Its symmetry implies the associated curious Poincaré duality, while the general mixed-Hodge-polynomial identity remains unresolved in the source.