Invertibility conjecture for the left partially dualized quasi-Hopf algebra

Let $H$ be a Hopf algebra, let $B$ be a left coideal subalgebra, and let $(H/B^+H)^\ast\#B$ be the associated left partially dualized quasi-Hopf algebra. In Remark on invertibility, the equivalent properties are that the image of the unit under the preantipode is invertible, that an antipode has both invertible distinguished elements, and that every antipode has both invertible distinguished elements. Invertibility conjecture. These equivalent properties always hold for the left partially dualized quasi-Hopf algebra

$$(H/B^+H)^\ast\#B.$$

The conjecture concerns whether the preantipode of every such left partially dualized quasi-Hopf algebra has invertible image of the unit, equivalently whether its antipodes have invertible distinguished elements. The source does not determine whether this is true.