Uniqueness of maximizer of dual Kantorovich problem with quadratic(or any strictly convex) cost

Question

I am considering the optimal transport problem under the setting $X=\mathbb{R}^n$, $\mu,\nu\in\mathcal{P}(X)$ be two probability measures, and the cost function is $c(x,y)=|x-y|^2$. We know from Brenier's theorem that when $\mu\ll Lebesgue$, then there exists a unique transport map $T=x-\nabla\phi$ optimizes the Kantorovich problem, and that $\phi$, together with its $c-$transform $\phi^c(y)=\inf_{x}\{c(x,y)-\phi(x)\}$, solves the dual Kantorovich problem: $$(\phi(x),\phi^c(y))=argmin\int\phi(x)d\mu(x)+\int\phi^c(y)d\nu(y),$$

among all functions $\phi$ such that $\phi(x)+\phi^c(y)\leq c(x,y)$.

My question is that under this condition(Euclidean space with quadratic cost, and $\mu\ll Lebesgue$), whether $\phi$ is the unique maximizer of the dual problem or not? Or in other words, for any maximizer $\phi$ of the dual problem, is it true or false that $(x-\nabla\phi)$ is an optimal transport map?

Answer 1

There exists some recent work dealing with it, see Lemma 2.1 in https://arxiv.org/abs/1705.01299 for the quadratic case and theorem 2.6 in https://arxiv.org/abs/2102.06379 for the general one.