Additivity–operator convexity conjecture for unitarily covariant channels

Let $\mathcal{U}$ be the set of functions that are additive for every pair of unitarily covariant quantum channels $(\Phi,\Omega)$. A function is operator convex when it satisfies the operator-convexity condition on an interval $I\subset\mathbb{R}$. The set $\mathcal{U}$ is a closed convex cone. Additivity–operator convexity conjecture. There exists an interval $I\subset\mathbb{R}$ such that $\mathcal{U}$ agrees with the set of operator convex functions on $I$. Determining $\mathcal{U}$ explicitly is presented as an interesting problem; the conjecture is motivated by the known Werner–Holevo result and remains open.