The moduli spaces of one-dimensional sheaves on P^2, first studied by Simpson and Le Potier, admit a Hilbert-Chow morphism to a projective base that behaves like a completely integrable system. Following a proposal of Maulik-Toda, one expects to obtain certain BPS invariants from the perverse filtration on cohomology induced by this morphism. This motivates us to study the cohomology ring structure of these moduli spaces. In this talk, we present a minimal set of tautological generators for the cohomology ring, and propose a “Perverse = Chern” conjecture concerning these generators, which specializes to an asymptotic product formula for refined BPS invariants of local P^2. This can be viewed as an analogue of the recently proved P=W conjecture for Hitchin systems. Based on joint work with Junliang Shen, and with Yakov Kononov and Junliang Shen.