Hausel–Letellier–Villegas purity conjecture for character varieties

Let $\M^\mu_\B$ be the smooth character variety associated with generic semisimple conjugacy classes of parabolic type $\mu$, let $\bar{\M}^\mu_\B$ be its corresponding $\PGL_n$ quotient, and let $(\Gamma,\vv)$ be the star-shaped (crab-shaped) quiver and dimension vector determined by $\mu$. Write $A_\Gamma(\vv,q)$ for the polynomial counting absolutely indecomposable representations of dimension $\vv$ over finite fields, let $PH(\bar{\M}^\mu_\B,x,y)$ denote the pure part of the cohomology, and let $d_\mu$ be the dimension of $\M^\mu_\B$. Hausel--Letellier--Villegas' purity conjecture.

$$PH(\bar{\M}^\mu_\B,x,y)=(xy)^{d_\mu/2}A_\Gamma(\vv,1/(xy)).$$

The conjecture gives a cohomological interpretation of the Kac polynomial and implies non-negativity of its coefficients for crab-shaped quivers. The text says it follows from a later master conjecture when $\mu$ is indivisible, but gives no resolution in full generality; its status is therefore open.