Let $n\in\mathbb N$, let $\mathcal P_2(\mathbb R^n)$ denote the set of Borel probability measures on $\mathbb R^n$ with finite second moment, and let $U:[0,\infty)\to\mathbb R$ be a convex function in the McCann displacement convexity class $\mathcal{DC}_n$, meaning that the McCann transform
Assume that $T:\mathbb R^n\to\mathbb R^n$ is the Brenier optimal transport map pushing $\mu_0$ forward to $\mu_1$, that $T$ is continuously differentiable, and that its Jacobian matrix $JT_x$ is symmetric positive semidefinite for $\mathcal L^n$-a.e. $x\in\mathbb R^n$. For each $t\in[0,1]$, define
and assume that $\mu_t:=(T_t)_\#\mu_0$ is absolutely continuous with density $\rho_t:\mathbb R^n\to[0,\infty)$. Assume also that there is a full-$\mathcal L^n$-measure Borel set $E\subset\mathbb R^n$ such that, for every $t\in[0,1]$, $JT_{t,x}:=(1-t)I_n+tJT_x$ exists on $E$, $\det JT_{t,x}>0$ on $E$, the Monge-Ampere change-of-variables identity
holds for every nonnegative Borel function $h:\mathbb R^n\to[0,\infty]$ and every Borel function $h:\mathbb R^n\to\mathbb R$ for which one side is absolutely integrable. For an absolutely continuous probability measure $\mu=\rho\,\mathcal L^n$, define
with values in $(-\infty,+\infty]$ whenever the integral is well-defined. Assume that $\mathcal U[\mu_0]$ and $\mathcal U[\mu_1]$ are not both involved in an indeterminate extended-real expression on the right-hand side below. Then, for every $t\in[0,1]$,