Kononov–Pi–Shen's P=C conjecture for moduli spaces of sheaves on the plane

Let $H$ be the hyperplane class in $\mathbb{P}^2$. For coprime $(d,\chi)$, let $M(d,\chi)$ be the smooth projective moduli space of stable sheaves supported on curves in $|dH|$ and having Euler characteristic $\chi$. Write $P_kH^{*\leq 2d-4}(M(d,\chi),\mathbb{Q})$ for the perverse filtration induced by the relevant moduli-space fibration, and let $C_kH^{*\leq 2d-4}(M(d,\chi),\mathbb{Q})$ be the span of monomials in the generators $c_{k_i}(j_i)$ with $\sum_i k_i\leq k$.

Kononov–Pi–Shen's P=C conjecture. For $d\geq 3$, we have

$$P_kH^{*\leq 2d-4}(M(d,\chi),\mathbb{Q})=C_kH^{*\leq 2d-4}(M(d,\chi),\mathbb{Q}).$$

This conjecture proposes that the perverse filtration agrees with the Chern filtration on the free part of the cohomology of these moduli spaces, providing an analogue of the P=W phenomenon even though the Hilbert–Chow morphism for $\mathbb{P}^2$ is not Lagrangian. Its resolution status is not specified in the supplied text.