The $P=C$ conjecture for the free cohomology of local $\mathbb{P}^2$ moduli spaces

Let $M_{d,\chi}$ be the moduli space considered in the paper, with perverse filtration $P_k$ on its cohomology. For $d\geq 3$, let $H^{*\leq 2d-4}(M_{d,\chi},\mathbb{Q})$ be the free part of the cohomology, and let $C_k$ be the Chern filtration defined by tautological classes. The $P=C$ conjecture. For $d\geq 3$, we have

$$P_k H^{*\leq 2d-4}(M_{d,\chi},\mathbb{Q})=C_k H^{*\leq 2d-4}(M_{d,\chi},\mathbb{Q}).$$

This conjecture proposes a cohomological lift of the refined BPS stabilization statement, identifying the perverse and Chern filtrations on the free part. It is proved for degrees $d=3,4$ in the paper, while the general case remains open.