3 problems

Equality conjecture. For each fixed $g\in{\cal G}$,

Let $\Phi$ be a quantum channel, let $\kappa\in\mathcal{K}$ be a symmetric operator-monotone metric function, and let $g\in\mathcal{G}_{\rm sym}$ be the corresponding symmetric rel…

Optimality conjecture. Equality holds in the three displayed bounds above.