General theory of Relativity: does a rapidly rotating body fall slower than a non-rotating body?

Does a rapidly rotating body fall slower than a non-rotating body? This difficult question requires the choice of suitable relativistic generalizations. The vertical free fall of a rotating dumb-bell in a Schwarzschild field and that of an horizontal rotating ring are both integrable. Their differences with the free fall of non-rotating bodies are extremely small. Notice that this effect differs from the precession of their spin