  We prove that the intersection cohomology (together with the perverse and the
Hodge filtrations) for the moduli space of one-dimensional semistable sheaves
supported in an ample curve class on a toric del Pezzo surface is independent
of the Euler characteristic of the sheaves. We also prove an analogous result
for the moduli space of semistable Higgs bundles with respect to an effective
divisor $D$ of degree $\mathrm{deg}(D)>2g-2$. Our results confirm the
cohomological $\chi$-independence conjecture by Bousseau for $\mathbb{P}^2$,
and verify Toda's conjecture for Gopakumar-Vafa invariants for certain local
curves and local surfaces.
  For the proof, we combine a generalized version of Ng\^o's support theorem, a
dimension estimate for the stacky Hilbert-Chow morphism, and a splitting
theorem for the morphism from the moduli stack to the good GIT quotient.
