The P=C conjecture for the projective plane

Let $M_{d,\chi}$ be the moduli space associated with the curve class $\beta=dH$ and Euler characteristic $\chi\in\mathbb{Z}$ on $\mathbb{P}^2$, where $H$ is the class of a line and $\operatorname{gcd}(d,\chi)=1$. Let $P_k H^*(M_{d,\chi})$ and $C_k H^*(M_{d,\chi})$ denote the perverse and Chern filtrations, respectively. The $P=C$ conjecture. The two filtrations agree on total cohomology:

$$P_k H^{*}(M_{d,\chi})=C_k H^{*}(M_{d,\chi}).$$

This is the del Pezzo analogue for $\mathbb{P}^2$ of the $P=W$ conjecture in non-abelian Hodge theory. The source recalls this conjecture but does not state a resolution of it here.