Skryabin's conjecture on bijective antipodes of noetherian Hopf algebras

Let $H$ be a noetherian Hopf algebra, with antipode $S$.

Skryabin's conjecture. The antipode $S$ is bijective.

This conjecture concerns whether the finite-dimensional result that Hopf algebras have bijective antipodes extends to the noetherian infinite-dimensional setting. Skryabin proved injectivity under these hypotheses, but surjectivity is not established in the statement given here.