Symmetry of eigenvalues of operators associated with representations of compact quantum groups

We ask the question whether for a given unitary representation $U$ the associated operator $\rho_{U}\in\operatorname{Mor}(U,U^{c\, c})$ has spectrum invariant under inversion -- in this case we say that $\rho_{U}$ has symmetric eigenvalues. This does not have to be the case. However, we give affirmative answer whenever a certain condition on the growth of dimensions of irreducible subrepresentations of tensor powers of $U$ is imposed. This condition is satisfied whenever $\widehat{\mathbb{G}}$ is of subexponential growth.