  We give a formula comparing the E-series of the moduli stacks of rank 2
degree 0 semistable Higgs bundles in genus $g \geq 2$ to intersection
E-polynomials of its coarse moduli space. A parellel formula holds in various
2-Calabi-Yau settings, for example for sheaves on K3 surfaces, or preprojective
algebras of $g$-loop quivers. As a consequence we provide evidence for a
conjecture of Davison on the BPS cohomology of Higgs bundles, which has
implications for non-abelian Hodge theory for stacks. We apply the formula to
cohomological $\chi$-independence tests for BPS cohomology of Higgs bundles and
K3 surfaces.
