Marginals with finite repulsive cost

We consider a multimarginal transport problem with repulsive cost, where the marginals are all equal to a fixed probability $\rho \in \mathcal{P}(\mathbb{R}^d)$. We prove that, if the concentration of $\rho$ is less than $1/N$, then the problem has a solution of finite cost. The result is sharp, in the sense that there exists $\rho$ with concentration $1/N$ for which $C(\rho) = \infty$.