When does Morita equivalence between two Hopf-von Neumann algebras imply also equivalence of their categories of comodules?

Let $A$ and $B$ be two Hopf-von Neumann (bi)algebras. Furthermore, let us assume that we know that they are Morita equivalent as von Neumann algebras (i.e. their categories of appropriate representations are equivalent). Under what additional assumption can we know that the categories of their comodules are also equivalent? What could be said about the relationship between these comodule categories in a generic case?