Let $V$ be a (right-)$H$ comodule wrt a coaction $\Delta_R$, where $H$ is a co-quasi-triangular Hopf algebra with co-quasi-triangular Hopf algebra structure $R$. It is well-known that $V$ has a braiding $$ \sigma : V \otimes V \to V \otimes V, ~~~~ v \otimes w \mapsto w^{(0)} \otimes v^{(0)} R(v^{(1)}\otimes w^{(1)}), $$ that commutes (of course) with the tensor product coaction; that is, $$ \Delta_R \otimes \Delta_R \circ \sigma = (\sigma \otimes \text{id} ) \circ (\Delta_R \otimes \Delta_R). $$

Do there exist any other such braidings for $V$?