Numerical $P=W$ conjecture for the Hitchin fibration

Let ${\mathcal M}_{\mathrm{Dol}}$ be the moduli space of rank $n$ Higgs bundles of degree $1$, let ${\mathcal M}_B$ be the corresponding character variety, and let ${}^\mathfrak{p}h^{i,j}({\mathcal M}_{\mathrm{Dol}})$ denote the perverse numbers associated with the Hitchin fibration. Let $h^{i,j}({\mathcal M}_B)$ be the Hodge numbers of the character variety, defined by

$$h^{i,j}({\mathcal M}_B)=\operatorname{dim}\left({}^j\mathrm{Hdg}^{i+j}({\mathcal M}_B)\right).$$

Numerical $P=W$ conjecture. The perverse numbers of the Hitchin fibration and the Hodge numbers of the corresponding character variety should agree:

$${}^\mathfrak{p}h^{i,j}({\mathcal M}_{\mathrm{Dol}})=h^{i,j}({\mathcal M}_B).$$

This is the numerical consequence of the expected identification of the perverse filtration on the Dolbeault moduli space with the weight filtration on the character variety under Simpson's nonabelian Hodge correspondence. The broader $P=W$ phenomenon has been established in some cases and has substantial partial progress, but is not resolved in general.