Purity conjecture for resolved Coulomb branches of connected semisimple groups

The purpose of this conjecture is to extend the purity phenomenon established here for the universal centralizer $J_G$ when $G$ is of adjoint form to Coulomb branches associated with more general connected semisimple groups. A resolved Coulomb branch means a resolved Coulomb branch associated with a connected semisimple group in the sense of Braverman–Finkelberg–Nakajima.

Purity conjecture. The rational cohomology of every resolved Coulomb branch associated with connected semisimple groups, in the sense of Braverman–Finkelberg–Nakajima, has pure Hodge structure.

This conjecture is motivated by analogies between Coulomb branches and Hitchin systems. The paper proves the corresponding statement for the universal centralizer $J_G$ when $G$ is of adjoint form, while the conjecture for all such resolved Coulomb branches remains open.