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The moduli spaces of one-dimensional sheaves on P^2 are first studied by Simpson and Le Potier, and they 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, which motivates us to study the cohomology ring structure of these moduli spaces. In this talk, we present some recent progress on this cohomology ring, including a minimal set of tautological generators, and a “Perverse = Chern” conjecture 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.