Absense of loops for the Wasserstein-$\mathcal{H}^1$ problem: the localization/blow-up argument

In the present work we prove that minimizers of the Wasserstein-$\mathscr{H}^1$ problem, introduced recently by Chambolle et. al., are trees in two cases: when the target measure is a sum of finitely many Dirac masses or when it has a bounded density.