When considering a single photon the classical explanation certainly fails. However, this is different from the claim that an effect perceived at macro levels c

When considering a single photon the classical explanation certainly fails. However, this is different from the claim that an effect perceived at macro levels cannot be explained by classical theory and only by quantum mechanics. The paper's result sounds to be along similar lines - claiming that a certain number "could only have come from quantum theory" while classical physics perfectly well yields the same number. By virtue of being a super-theory of classical mechanics, all phenomena are quantum phenomena with quantum explanations, so that's not what is being talked about here.