The cohomological P=W conjecture

Let $C$ be a punctured Riemann surface, let $G$ be a linear reductive algebraic group over $\mathbb{C}$, and let $\mathcal{M}_B$ and $\mathcal{M}_{Dol}$ be corresponding Betti and Dolbeault moduli spaces under the non-abelian Hodge correspondence $\phi$. Let $W_\bullet$ denote the weight filtration on $H^*(\mathcal{M}_B)$, and let $P_\bullet$ denote the Perverse-Leray filtration on $H^*(\mathcal{M}_{Dol})$ associated to the Hitchin map $h:\mathcal{M}_{Dol}\to\mathbb{A}$. Cohomological P=W conjecture. Under $\phi$, the weight and perverse filtrations coincide:

$$\phi^*(W_{2k}H^*(\mathcal{M}_B)=W_{2k+1}H^*(\mathcal{M}_B))=P_kH^*(\mathcal{M}_{Dol}).$$

The conjecture has been resolved for rank two over Riemann surfaces of any genus and for arbitrary rank over a genus-two Riemann surface. The general case remains open.