That is referenced in both the paper and the video in fact. Apparently Diaconis presented a model which predicted about 51% preference for "Same side" and also did 2500 flips and said that about 250k flips would be needed to get 3 sigma of significance. So this paper decided to test it empirically and got to about exactly that number after 350k flips from a team of researchers.

as long as a machine is used for consistency*