P=W conjecture for resolutions of singular character varieties

Let $X$ be a compact Riemann surface, let $G$ be a complex reductive algebraic group, and choose resolutions of singularities

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

Let $\widetilde{\Psi}\colon \widetilde{M}_{\mathrm{Dol}}(X,G)\to \widetilde{M}_{\mathrm{B}}(X,G)$ be the diffeomorphism lifting the non-abelian Hodge correspondence, and let $P_\bullet$ and $W_\bullet$ denote the perverse and weight filtrations, respectively. P=W conjecture for resolution. For every $k$,

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

The conjecture concerns replacing singular or intersection cohomology by cohomology of resolutions. The paper relates it to the PI=WI conjecture through the decomposition theorem, but no general resolution-level result is supplied here.