P=W for resolutions of character varieties

Let $C$ be a smooth projective curve and $G$ a reductive group. Let $M_{\mathrm{Dol}}(C,G)$ and $M_{\mathrm{B}}(C,G)$ be the untwisted Dolbeault and Betti moduli spaces, with non-abelian Hodge correspondence $\Psi$. The conjecture asks for resolutions of singularities

$$f_{\mathrm{Dol}}\colon \widetilde{M}_{\mathrm{Dol}}(C,G)\to M_{\mathrm{Dol}}(C,G),\qquad f_{\mathrm{B}}\colon \widetilde{M}_{\mathrm{B}}(C,G)\to M_{\mathrm{B}}(C,G),$$

together with a diffeomorphism $\widetilde{\Psi}\colon\widetilde{M}_{\mathrm{Dol}}(C,G)\to\widetilde{M}_{\mathrm{B}}(C,G)$ lifting $\Psi$ and making the induced cohomological square commute.

P=W for resolution. The lift satisfies, for every $k$,

$$P_kH^*(\widetilde{M}_{\mathrm{Dol}}(C,G),\mathbb{Q})=\widetilde{\Psi}^*W_{2k}H^*(\widetilde{M}_{\mathrm{B}}(C,G),\mathbb{Q}).$$

This is a strong form of PI=WI. The source states that it was proved for character varieties admitting a symplectic resolution, but explains that it fails when no symplectic resolution exists; the unrestricted formulation is therefore refuted.