Metric spaces nonembeddable into Banach spaces with the Radon-Nikodym property and thick families of geodesics

We show that a geodesic metric space which does not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodym property does not necessarily contain a bilipschitz image of a thick family of geodesic's. This is done by showing that no thick family of geodesics is Markov convex, and comparing this result with results of Cheeger-Kleiner, Lee-Naor, and Li. The result contrasts with the earlier result of the author that any Banach space without the Radon-Nikodym property contains a bilipschitz image of a thick family of geodesics.