Uniform Bi-Hölder Transport from Weak MTW

We prove that the weak Ma--Trudinger--Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density regularity is assumed.