  We show continuity of the martingale optimal transport optimisation problem
as a functional of its marginals. This is achieved via an estimate on the
projection in the nested/causal Wasserstein distance of an arbitrary coupling
on to the set of martingale couplings with the same marginals. As a corollary
we obtain an independent proof of sufficiency of the monotonicity principle
established in [Beiglboeck, M., & Juillet, N. (2016). On a problem of optimal
transport under marginal martingale constraints. Ann. Probab., 44 (2016), no.
1, 42106] for cost functions of polynomial growth.
