On calculating the supremums of convex functions from output eigenvalues of operator

Let $\mathcal U$ be a finite subgroup of the group of all unitary operators in a Hilbert space $H$. If a map $\Phi $ on the convex set $\mathfrak {S}(H)$ consisting of positive operators with unit trace (quantum states) has the form
$$ \Phi (\rho )=\sum \limits _{U\in {\mathcal U}}\pi _UU\rho U^*, \ \rho \in \mathfrak {S}(H), $$
where $(\pi _U)$ is a probability distribution on $\mathcal U$, then it is said to be a mixed unitary channel. Denote $P_f$ the orthogonal projector on one-dimensional subspace $\{\mathbb {C}f\}$, where $f$ is a unit vector $H$. Given a convex function of one variable $F(x),\ 0\le x\le 1$, consider the task of calculating the supremum across all $f$ of the quantity $\sum \limits _jF(\lambda _j)$, where $(\lambda _j)$ are eigenvalues of the operator $\Phi (P_f)$. The report will be. devoted to finding the conditions under which the supremum is achieved on elementary tensors $f=\bigotimes \limits _jf_j$ in the case if ${\mathcal U}=\bigotimes \limits_j{\mathcal U}_j$.