Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group. Consider the associated dense Hopf$^*$-subalgebra $\mathcal{O}(\mathbb{G})$ and let $S: \mathcal{O}(\mathbb{G})\to \mathcal{O}(\mathbb{G})$ denote its antipode. In the book "Compact quantum groups and their representation categories", it is shown that $S$ is unbounded when $\mathbb{G}$ is not of Kac-type. The converse is also true, namely if $\mathbb{G}$ is of Kac-type, then $S$ is bounded. To see this, note that the assumption implies that $S$ is $*$-preserving, and thus $S$ is a positive map because $$S(x^*x) = S(x)S(x^*) = S(x)S(x)^*\ge 0$$ for any $x \in \mathcal{O}(\mathbb{G})$. Note that $\mathcal{O}(\mathbb{G})$ is an operator system, and since a positive unital map on an operator system is necessarily bounded (See Paulsen's book "Completely bounded maps and operator algebras", proposition 2.1), we conclude that $S$ is bounded. Moreover, it also follows that $\|S\| \le 2\|S(1)\| = 2.$

**Question**: Can we say something more about the norm $\|S\|$? Is it possible that $\|S\| = 2?$