The P=W conjecture for non-abelian Hodge moduli spaces

Let $X$ be a smooth complex projective variety and $G$ an algebraic reductive group. Let $M_D$ be the moduli space of Higgs $G$-bundles, with proper Hitchin map, and let $M_B$ be the corresponding character variety. Non-abelian Hodge theory gives a canonical identification of their cohomology groups, carrying the perverse filtration $P$ on $H^*(M_D)$ and the mixed-Hodge-theoretic weight filtration $W$ on $H^*(M_B)$. The $P=W$ conjecture. Whenever non-abelian Hodge theory holds, the perverse filtration and the weight filtration correspond under this canonical identification. This extends the known $P=W$ result for rank-two Higgs bundles on curves of genus at least $2$ and predicts analogous statements for the many Hitchin-type moduli spaces arising in non-abelian Hodge theory.