De Cataldo–Hausel–Migliorini's P=W conjecture

Let $\mathcal{M}_{B}(r,n)$ and $\mathcal{M}_{Dol}(r,n)$ be the Betti and Dolbeault moduli spaces, let $h$ be the Hitchin map, and let $W_\bullet$ and $P_\bullet$ denote the weight and perverse filtrations on their cohomology. For coprime integers $r,n$ with $r>0$, let

$$\psi:\mathcal{M}_{B}(r,n)\to\mathcal{M}_{Dol}(r,n)$$

be the real analytic nonabelian Hodge isomorphism. P=W conjecture. The induced cohomological isomorphism

$$\psi^*:H^*(\mathcal{M}_{Dol}(r,n),\mathbb{Q})\to H^*(\mathcal{M}_{B}(r,n),\mathbb{Q})$$

should satisfy, for every $k\in\mathbb{Z}$,

$$P_kH^*(\mathcal{M}_{Dol}(r,n),\mathbb{Q})\xrightarrow{\simeq}W_{2k}H^*(\mathcal{M}_{B}(r,n),\mathbb{Q}).$$

The conjecture would identify the perverse filtration arising from the Hitchin fibration with the weight filtration on the character variety, explaining the correspondence between relative and curious Hard Lefschetz. The source does not provide resolution evidence for this candidate.