The classification of finite-dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero and whose group of group-like elements is abelian is very much completed. However, from a more categorical point of view it would make sense to classify such Hopf algebras up to gauge equivalence,i.e. two (quasi-)Hopf algebras are called gauge equivalent if their representations categories are tensor equivalent.

Are people interested in the latter problem? Are there results in this direction?