Existence of a quasi-bialgebra with preantipode that is not quasi-Hopf

A quasi-bialgebra is an algebraic structure with coproduct, counit, and associator satisfying the quasi-bialgebra axioms; a preantipode is a map satisfying the three preantipode axioms, while a quasi-Hopf algebra is a quasi-bialgebra equipped with a quasi-antipode. Existence conjecture. There is a quasi-bialgebra with preantipode which is not a quasi-Hopf algebra.

Every quasi-Hopf algebra has a preantipode, and the associated adjunction is an equivalence of categories. The converse is asserted here as likely but remains open because the authors do not provide an example.