Let $n\in\mathbb N$, let $\tau>0$, and let $\mu_{\tau,k}\in\mathcal P_2(\mathbb R^n)$. Let $V\in C^1(\mathbb R^n)$ be bounded below, and assume that there is a constant $C>0$ such that
when $\nu=r\mathcal L^n$ for a Borel density $r:\mathbb R^n\to[0,\infty)$, the entropy integral is finite, and the potential integral is finite, and set $\mathcal F[\nu]=+\infty$ otherwise.
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Suppose that $\mu_{\tau,k+1}\in\mathcal P_2(\mathbb R^n)$ minimises
over $\mathcal P_2(\mathbb R^n)$, with finite value at $\mu_{\tau,k+1}$, and suppose that $\mu_{\tau,k+1}=\rho_{\tau,k+1}\mathcal L^n$, where $\rho_{\tau,k+1}\in C^1(\mathbb R^n)$ is strictly positive. If $T:\mathbb R^n\to\mathbb R^n$ is a Borel quadratic-cost optimal transport map satisfying