The P=W conjecture for Higgs and Betti moduli spaces

Let $G$) be a reductive group, and let $M_\mathrm{H}$ and $M_\mathrm{B}$ be the corresponding Hitchin and Betti moduli spaces for a curve $C$. Let $h:M_\mathrm{H}\to\mathbb{A}^d$ be the Hitchin map, and let $P_\bullet$ denote the perverse Leray filtration on $H^k(M_\mathrm{H};\mathbb{Q})$ induced by $Rh_*\mathbb{Q}_{M_\mathrm{H}}$. Let $W_\bullet$ denote the weight filtration on $H^k(M_\mathrm{B};\mathbb{Q})$. The P=W conjecture. For all $j$ and $k$,

$$W_{2j-2}H^k(M_\mathrm{B};\mathbb{Q})=W_{2j-1}H^k(M_\mathrm{B};\mathbb{Q})=P_{j-1}H^k(M_\mathrm{H};\mathbb{Q}).$$

This predicts that the weight filtration on the Betti moduli space agrees with the perverse Leray filtration associated with the Hitchin fibration. It has been proved for $G=\mathrm{SL}_2(\mathbb{C})$, $\mathrm{GL}_2(\mathbb{C})$, and $\mathrm{PGL}_2(\mathbb{C})$, but remains open in general.