Consider a coquasitriangular Hopf-algebra $(H,\mu,\eta,\Delta,\epsilon, S;r)$ over a field $\mathbb F$ with characteristic zero and the braided monoidal category $\mathcal M^H$ of $H$-right-comodules.
Let $B:=(H,\overline \mu,\eta, \Delta,\epsilon, \overline S)$ with \begin{align*} \overline \mu &: H \otimes_\mathcal C H \rightarrow H , h\otimes g \mapsto h_{(2)}g_{(2)} r(S(h_{(1)})h_{(3)} \otimes S(g_{(1)}))\\ \overline S &: H \rightarrow H , S(h_{(2)})r(S^2(h_{(3)})S(h_{(1)}) \otimes h_{(4)}) \end{align*} be the (transmutated) Hopf algebra object in $\mathcal M^H$ 'corresponding' to $H$.
That is $B$ is as $H$-right comodule given by \begin{align*} \delta: H \rightarrow H \otimes_\mathbb F H, h \mapsto h_{(2)}\otimes S(h_{(1)})h_{(3)} \end{align*} and $\mu,\eta, \Delta,\epsilon, \overline S$ are $H$-right comodule morphisms such that the defining axioms of a Hopf algebra are satisfied.
Is the antipode $\overline S$ invertible?