P=W conjecture for twisted character varieties

Let $C$ be a smooth projective curve, let $G$ be one of $\operatorname{GL}_n$, $\operatorname{SL}_n$, or $\operatorname{PGL}_n$, and let $d$ satisfy $\mathrm{gcd}(n,d)=1$. Define the twisted character variety $M^{\mathrm{tw}}_{\mathrm{B}}(C,G,d)$ and the corresponding Dolbeault moduli space $M^{\mathrm{tw}}_{\mathrm{Dol}}(C,G,d)$, with non-abelian Hodge diffeomorphism $\Psi\colon M^{\mathrm{tw}}_{\mathrm{Dol}}(C,G,d)\to M^{\mathrm{tw}}_{\mathrm{B}}(C,G,d)$. Let $P_k$ denote the perverse Leray filtration associated with the Hitchin fibration on the Dolbeault side, and let $W_{2k}$ denote the weight filtration on the Betti-side cohomology.

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

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

The conjecture predicts that non-abelian Hodge theory identifies filtrations of different geometric origins. It was verified in rank $2$, while the general twisted case remains open.