  We explore connections between three structures associated with the
cohomology of the moduli of 1-dimensional stable sheaves on $\mathbb{P}^2$:
perverse filtrations, tautological classes, and refined BPS invariants for
local $\mathbb{P}^2$. We formulate the $P=C$ conjecture identifying the
perverse filtration with the Chern filtration for the free part of the
cohomology. This can be viewed as an analog of de Cataldo--Hausel--Migliorini's
$P=W$ conjecture for Hitchin systems. Our conjecture is compatible with the
enumerative invariants of local $\mathbb{P}^2$ calculated by refined
Pandharipande--Thomas theory or Nekrasov partition functions. It provides a
cohomological lift of a conjectural product formula of the asymptotic refined
BPS invariants. We prove the $P=C$ conjecture for degrees $\leq 4$.
