Left–right symmetry of faithful flatness over right coideal subalgebras

Ascertain whether faithful flatness of a Hopf algebra H with bijective antipode over a right coideal subalgebra A is a left–right symmetric property; specifically, prove or refute that left faithful flatness of H over A implies right faithful flatness over A (and vice versa) in general.

Background

Masuoka asked whether, under left faithful flatness of H over a right coideal subalgebra A, every nonzero object of AMH is a projective generator in AM. The authors observe (Lemma 8.1) that if all objects of MH are flat A-modules, then H is faithfully flat over A on both sides, suggesting a symmetry phenomenon.

They explicitly raise and leave open the broader question of whether faithful flatness is inherently left–right symmetric for arbitrary right coideal subalgebras, noting it is known in special cases (e.g., when A is a subbialgebra or under Theorem 7.4’s hypotheses).