Everything is finite-dimensional over a field $k$.
Let $B$ be a bialgebra with $B\text{-mod}$ its category of modules. Suppose now that we fix a rigid structure on $B\text{-mod}$ such that the dual of a module is given by the the dual vector space, with some action. In other words, taking duals in $B\text{-mod}$ commutes on the nose with the canonical forgetful functor to $Vect$.
My question: Is $B$ necessarily a Hopf algebra?
The reason for this question is that from the above data I can easily define a quasi-Hopf algebra structure on $B$, whose coassociator is trivial. A quasi-Hopf algebra for me is like a Hopf algebra, except that its not quite coassociative and that the antipode has been replaced by a triple $(S,\alpha, \beta)$, where $S$ is still the antipode, but $\alpha, \beta \in H$ are now some elements implementing the evaluation and coevaluation in $B\text{-mod}$. This triple has to satisfy some axioms, namely the zig-zag equations of the rigid structure. ${}^\star$
Thus
My question formulated differently: Is it necessarily the case that $\alpha = \beta = 1$ in the quasi-Hopf algebra I obtained above?
${}^\star$ A source for this construction is eg Section 3.5 in "Quasi-Hopf algebras - a categorical approach" by Bulacu, Caenepeel, Panaite, and van Oystaeyen.