Assume that I have a pure, multipartite state $\omega_{ABCD}$ and a unitary $U(\omega_{ABCD})=\tau_{ABCD}$. The effect of $U$ on $\omega$ results in$$S(\omega_B)<S(\tau_B) $$ and$$S(\omega_A)>S(\tau_A).$$$S$ is the von Neumann entropy and the states $\omega_A,\tau_A$ etc are the marginal density matrices (for instance, $\omega_A$= $Tr_{BCD}{\omega_{ABCD}}$). This effect of $U$ is an assumption, I don't say it always holds.QUESTION:Comparing the joint entropies $S(\omega_{AB})$ and $S(\tau_{AB})$, is this true$$S(\tau_{AB})-S(\omega_{AB})\leq S(\tau_{B})-S(\omega_{B})$$and if yes, why?EDIT: Another assumption is that $S(\omega_{AB})\geq S(\omega_B)$.
Entropy inequality of joint von Neumann entropy given some marginal inequalities
quantum information