Displacement Convexity of Minus the Free Entropy: the Tangent Inequality along Optimal Couplings

In the setting of Free Products, Semicircular Laws, the Free Heat Flow, Free Fisher Information and Free Entropy: Standing Notation, the conjugate variables of a law $\mu$ that has them are written $\xi_{\mu}$, and $\mathcal{J}^{1}_{\gamma}$ is the coupling pairing.

(Tangent inequality)¶ The pair $(\mathcal{D}_{\chi},-\chi^{*})$ satisfies the tangent inequality clause of a free entropy penalty: for every $\mu\in\mathcal{D}_{\chi}$ that has conjugate variables, every $\nu\in\mathcal{D}_{\chi}$ and every optimal coupling $\gamma\in\Pi(\mu,\nu)$,

$$-\chi^{*}(\nu)\ge-\chi^{*}(\mu)+\mathcal{J}^{1}_{\gamma}(\xi_{\mu}).$$