A p-value of 0.3% is not that low given the multiple hypothesis testing nature of the show.

A quick Google result shows that 592 total episodes have been filmed of the show. Under the null hypothesis that there was no cheating in any of the shows, you would still expect more likely than the chance that some show has a strong p-value of 0.3% for cheating to be 1-.997^592 = 83%.

As a jury, I don't think I would convict with just a p-value of 0.3% alone. You'd want something like a DNA test with a p-value expressed in 10^-6 or something to reduce the problem of multiple hypothesis testing / setup assumptions. Or you'd want to rely on other evidence like a confession or wiretap.