The $P=C$ conjecture for moduli spaces of one-dimensional sheaves

Let $M_{d,c7}$ be the moduli space of one-dimensional sheaves on $\mathbb{P}^2$ with degree $d$ and Euler characteristic $\chi$, and let $P_\bullet H^*(M_{d,\chi},\mathbb{Q})$ and $C_\bullet H^*(M_{d,\chi},\mathbb{Q})$ denote its perverse and Chern filtrations, respectively. The $P=C$ conjecture. For coprime $d\geq 1$ and $\chi\in\mathbb{Z}$, we have

$$P_\bullet H^*(M_{d,\chi}, \mathbb{Q}) = C_\bullet H^*(M_{d,\chi}, \mathbb{Q}).$$

This strengthens the previously proposed version, which asserted equality only up to cohomological degree $2d-4$. The conjecture relates the perverse filtration of the Hilbert--Chow map to the multiplicative filtration generated by normalized tautological Chern classes; the supplied text gives no resolution evidence.