We use a Korevaar-style maximum principle approach to show the following: Fixing a  bound on the log densities of a set of smooth measures, there is a quantifiably-sized Wasserstein neighborhood over which all pairs of such measures will enjoy smooth optimal transport. We do this in spite of unhelpful MTW curvature, by showing that when the gradient of the Kantorovich potential is small enough, the Hessian ``bound" places the Hessian in one of two disconnected regions, one bounded and the other unbounded.  Tracking the estimate along a continuity path which starts in the bounded region, we conclude the Hessian must stay bounded.