The P=W compatibility conjecture for nonabelian Hodge theory

Let $C$ be a smooth projective curve with underlying Riemann surface $\Sigma_g$. Let $\mathcal{H}^{\operatorname{Betti}}$ and $\mathcal{H}^{\operatorname{Dol}}$ be the graded BPS cohomological spaces of Betti representations and degree-zero Dolbeault Higgs bundles, respectively, and let $\Lambda:\mathcal{H}^{\operatorname{Betti}}\cong\mathcal{H}^{\operatorname{Dol}}$ be the graded vector-space isomorphism induced by nonabelian Hodge theory. P=W compatibility conjecture. The map $\Lambda$ takes the weight filtration on $\mathcal{H}^{\operatorname{Betti}}$ to a doubled perverse filtration on $\mathcal{H}^{\operatorname{Dol}}$. The conjecture refines the nonabelian Hodge correspondence by asserting compatibility between the weight filtration on the Betti side and the perverse filtration arising from the Hitchin system; the ordinary P=W conjecture mentioned in the source is already a theorem.