Intersection curious hard Lefschetz conjecture for character varieties

Let $C$ be a smooth projective curve and $G$ a reductive group. Let $M_{\mathrm{B}}(C,G)$ be the untwisted character variety, let $IH^*(M_{\mathrm{B}}(C,G))$ denote its intersection cohomology, and let $W$ be its weight filtration.

Intersection curious hard Lefschetz conjecture. There exists a class $\alpha\in H^2(M_{\mathrm{B}}(C,G))$ such that, for every $k$, cup product induces isomorphisms

$$\cup\alpha^k\colon \operatorname{Gr}^W_{\dim M_{\mathrm{B}}-2k}IH^*(M_{\mathrm{B}}(C,G))\xrightarrow{\simeq}\operatorname{Gr}^W_{\dim M_{\mathrm{B}}+2k}IH^{*+2k}(M_{\mathrm{B}}(C,G)).$$

In particular, the intersection E-polynomial of $M_{\mathrm{B}}(C,G)$ is palindromic.

This conjecture is presented as a consequence of PI=WI together with relative hard Lefschetz, and its resolution status is not given in the supplied text.