Equivalent version of the P=W conjecture with tautological and multiplicative splitting

Let $C$ be an irreducible nonsingular projective curve of genus $g\geq 2$, let ${\mathcal M}_{\mathrm{Dol}}$ be its twisted Dolbeault moduli space with Hitchin fibration $h$, and let $c(\gamma,k)$ denote the tautological classes. A splitting $G_*H^*({\mathcal M}_{\mathrm{Dol}},{\mathbb Q})$ of the perverse filtration is a decomposition whose degree-$k$ part is denoted $G_k$. Equivalent version of P=W. There exists a splitting $G_*H^*({\mathcal M}_{\mathrm{Dol}},{\mathbb Q})$ satisfying:

1. Every tautological class has perversity $k$:

$$c(\gamma,k)\in G_kH^*({\mathcal M}_{\mathrm{Dol}},{\mathbb Q}),\qquad \forall k\geq 0,\ \forall\gamma\in H^*(C,{\mathbb Q}).$$

2. The perverse decomposition is multiplicative:

$$\cup:G_kH^d({\mathcal M}_{\mathrm{Dol}},{\mathbb Q})\times G_{k'}H^{d'}({\mathcal M}_{\mathrm{Dol}},{\mathbb Q})\longrightarrow G_{k+k'}H^{d+d'}({\mathcal M}_{\mathrm{Dol}},{\mathbb Q}).$$

This criterion expresses the P=W conjecture through a splitting compatible with tautological generators and cup products. The supplied text does not establish whether this equivalent formulation is open or resolved.