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Einstein's theory of gravity is based on the Strong Equivalence Principle, which predicts that objects should fall the same way no matter how compact they are. Almost all other theories of gravity predict that very compact objects should begin to deviate. In this new system, we have the extremely compact pulsar and the less compact inner white dwarf falling in the gravity of the outer white dwarf. If they fall differently, we should see it in our measurements. If they fall in exactly the same way, then theorists will need more ingenuity to come up with new theories of gravity that can match Einstein's in this orbiting laboratory.
The discovery and timing of this new system are described in Ransom et al. 2014 (arXiv version).