Experiments so far have ruled out locality and realism as a combination. But could a theory assuming only one of them be correct" Nonlocal hidden variables (NLHV) theories would allow for the possibility of hidden variables but would concede nonlocality, the idea that a measurement on a particle at one location may have an immediate effect on a particle at a separate location.
Measuring the polarizations of the pairs of entangled particles in their setup, the researchers showed that the results did not agree with the predictions of certain NLHV theories but did agree with the predictions of quantum mechanics. In this way, they were able to rule out certain NLHV theories. Their results agree with other groups that have performed similar experiments.
I may have missed it, but I don't recall ever seeing any experiment on entanglement that hasn't produced any result that's consistent with QM. One can argue that such-and-such an experiment doesn't rule out that and that theory, but QM is batting with 100% hits here with zero strike-out. I find that rather impressive, and impressively convincing.