P=W conjecture for untwisted character varieties

Let $C$ be a smooth projective curve and $G$ a reductive group. Let $M_{\mathrm{B}}(C,G)$ and $M_{\mathrm{Dol}}(C,G)$ be the untwisted Betti and Dolbeault moduli spaces, respectively, and let $\Psi\colon M_{\mathrm{Dol}}(C,G)\to M_{\mathrm{B}}(C,G)$ be the non-abelian Hodge diffeomorphism. Let $P_k$ denote the perverse Leray filtration from the Hitchin fibration and $W_{2k}$ the weight filtration.

P=W conjecture for untwisted character varieties. For every $k$,

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

This conjecture asks for a P=W relation despite the failure of curious hard Lefschetz for ordinary cohomology in general. The supplied text gives no resolution of the conjecture.