  In this note, we complete the analysis of the Martingale Wasserstein
Inequality started in arXiv:2011.11599 by checking that this inequality fails
in dimension $d\ge 2$ when the integrability parameter $\rho$ belongs to
$[1,2)$ while a stronger Maximal Martingale Wasserstein Inequality holds
whatever the dimension $d$ when $\rho\ge 2$.
