Noetherian Hopf algebra domains of Gelfand–Kirillov dimension two

We classify all noetherian Hopf algebras H over an algebraically closed field k of characteristic zero which are integral domains of Gelfand–Kirillov dimension two and satisfy the condition . The latter condition is conjecturally redundant, as no examples are known (among noetherian Hopf algebra domains of GK-dimension two) where it fails.