P=C conjecture for one-dimensional sheaf moduli on del Pezzo surfaces

Let $S$ be a del Pezzo surface, let $H$ be the polarization, let $\beta$ be an effective divisor, let $\chi\in\mathbb{Z}$, and let $M_{\beta,\chi}$ be the moduli space of one-dimensional sheaves with determinant $\mathcal{O}_S(\beta)$ and Euler characteristic $\chi$. Assume that $\beta\cdot H$ and $\chi$ are coprime. Let $P_\bullet H^*(M_{\beta,\chi})$ be the perverse filtration associated with the Hilbert–Chow morphism, and let $C_\bullet H^*(M_{\beta,\chi})$ be the Chern filtration generated by the normalized tautological classes. P=C conjecture.

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

The identity relates the perverse and Chern filtrations and is intended to connect the geometry of these moduli spaces with enumerative invariants; the source states it as a conjecture for del Pezzo surfaces and provides evidence, but no resolution.