At one point, the article says "importantly, [the principle of diminishing returns] holds even along geodesics, making it distinct from and stronger than the triangle inequality." Later, they say "it is not trivial to verify whether any given path through color space is a geodesic. We chose the neutral axis because it is the one path on which all available data agree that it is indeed a geodesic", and go on to argue (if I am following it correctly) that it is unlikely that their principle of diminishing returns is just an artifact from this choice.
I do not know if the authors are claiming that what we perceive as color similarity is not a metric at all, but personally, I would not be surprised if it were not. See my other post for how my subjective perception of color differences seems to me.