Before revision:  For the latter, our main mathematical tool is a recent generalization of Strassen's theorem [S. Gerhold, I.C. G\"ul\"um , arXiv:1512.06640] , which characterizes the existence of martingales in balls w.r.t. the infinity Wasserstein distance .
After revision:  For the latter, our main mathematical tool is a recent result on approximation by peacocks [S. Gerhold, I.C. G\"ul\"um , arXiv:1512.06640] , which characterizes the existence of martingales in balls w.r.t. the infinity Wasserstein distance .