Exact catalytic entropy conjecture

Let $\rho$ and $\rho'$ be $d$-dimensional density matrices that are not unitarily equivalent. A finite-dimensional density matrix $\sigma$ and a unitary $U$ are sought such that

$$\operatorname{Tr}_1[U\rho\otimes\sigma U^\dagger]=\sigma,\qquad \operatorname{Tr}_2[U\rho\otimes\sigma U^\dagger]=\rho'.$$

Exact catalytic entropy conjecture. Such a density matrix $csigma$ and unitary $U$ exist if and only if $H(\rho)<H(\rho')$ and $\operatorname{rank}(\rho)\leq\operatorname{rank}(\rho')$. The paper presents an affirmative solution of this conjecture, establishing that the stated entropy and rank conditions are sufficient as well as necessary; it concerns exact catalytic transformations in quantum and classical probability settings.