  We show that a geodesic metric space which does not admit bilipschitz
embeddings into Banach spaces with the Radon-Nikod\'ym property does not
necessarily contain a bilipschitz image of a thick family of geodesics. This is
done by showing that any thick family of geodesics is not 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-Nikod\'ym property contains a bilipschitz image of a thick
family of geodesics.
