Curious hard Lefschetz conjecture for character varieties

Let $\mathcal{M}_\B$ be the relevant character variety, let $\dim=\dim(\mathcal{M}_\B)$, and let $\alpha\in W_4H^2(\mathcal{M}_\B)$, where $W_\bullet$ is the weight filtration. Curious hard Lefschetz conjecture. Cup product with powers of $\alpha$ induces isomorphisms

$$L^l:Gr^W_{\dim-2l}H^{i-l}(\mathcal{M}_\B)\xrightarrow{\cong}Gr^W_{\dim+2l}H^{i+l}(\mathcal{M}_\B),\qquad x\mapsto x\cup\alpha^l.$$

This is a hard-Lefschetz-type symmetry for the non-projective character variety, with a weight-four class replacing the usual weight-two ample class. The source reports a proof for $n=2$ and leaves the general conjecture open.