The Conjugate Variables of a Free Entropy Penalty are Displacement Monotone along Optimal Couplings

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $(\mathcal{D}_{0},\mathcal{E}_{0})$ be a free entropy penalty; the conjugate variables of a law $\lambda$ that has them are written $\xi_{\lambda}$, and the coupling pairings $\mathcal{J}^{1}_{\gamma}$ and $\mathcal{J}^{2}_{\gamma}$ are those of Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing.

¶If $\mu,\nu\in\mathcal{D}_{0}$ both have conjugate variables and $\gamma\in\Pi(\mu,\nu)$ is an optimal coupling, then

$$\mathcal{J}^{1}_{\gamma}(\xi_{\mu})-\mathcal{J}^{2}_{\gamma}(\xi_{\nu})\le0.$$