A complex (finite-dimensional) Hopf algebra is said to be a Kac algebra if it is a ${\rm C^{\star}}$-algebra in such a way that the comultiplication $\Delta$ is a $\star$-homomorphism. Obviously, a (finite-dimensional) Kac algebra is a semisimple Hopf algebra, but what about the converse:

Let $H$ be a complex finite dimensional semisimple Hopf algebra.
Question: Is there a Kac algebra $K$ isomorphic to $H$ as Hopf algebra?
If no, what is the smallest counter-example (for the dimension), and what is the main obstruction?