  We show the existence of a finite group $G$ having an irreducible character
$\chi$ with Frobenius-Schur indicator $\nu_2(\chi){=}{+}1$ such that $\chi^2$
has an irreducible constituent $\varphi$ with $\nu_2(\varphi){=}{-}1$. This
provides counterexamples to the positivity conjecture in rational CFT and a
conjecture of Zhenghan Wang about pivotal fusion categories.
