Sense of fairness: not by itself a moral sense and not a foundation of a lot of morality.
Baumard et al. make a good case that a sense of fairness evolved and that showing this requires reciprocity games with choice of partner. However, they oversimplify both morality and the evolution of morality. Where fairness is involved in morality, other things are, too, and fairness is often not involved. In the evolution of morality, other things played a role. Plus, the motive for being fair originally was self-interest, not anything moral.