PI=WI 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, $M_{\mathrm{Dol}}(C,G)$ its Dolbeault counterpart, and $\Psi\colon M_{\mathrm{Dol}}(C,G)\to M_{\mathrm{B}}(C,G)$ the non-abelian Hodge diffeomorphism. Write $IH^*$ for intersection cohomology, let $P_k$ be the perverse Leray filtration associated with the Hitchin fibration, and let $W_{2k}$ be the weight filtration.

PI=WI conjecture. For every $k$,

$$P_k IH^*(M_{\mathrm{Dol}}(C,G))=\Psi^*W_{2k}IH^*(M_{\mathrm{B}}(C,G)).$$

The conjecture was proposed to recover curious hard Lefschetz for the singular untwisted character variety; together with relative hard Lefschetz, it would imply intersection curious hard Lefschetz. Its status is not resolved in the supplied text.