  Stability of the value function and the set of minimizers w.r.t. the given
data is a desirable feature of optimal transport problems. For the classical
Kantorovich transport problem, stability is satisfied under mild assumptions
and in general frameworks such as the one of Polish spaces. However, for the
martingale transport problem several works based on different strategies
established stability results for R only. We show that the restriction to
dimension d = 1 is not accidental by presenting a sequence of marginal
distributions on R 2 for which the martingale optimal transport problem is
neither stable w.r.t. the value nor the set of minimizers. Our construction
adapts to any dimension d $\ge$ 2. For d $\ge$ 2 it also provides a
contradiction to the martingale Wasserstein inequality established by Jourdain
and Margheriti in d = 1.
