I don't think the JREF would accept them as an applicant. The JREF specializes on much smaller scale tests where they can stop shenanigans and allow for very low p-values. For example, they'll task a dowser with determining which of ten boxes filled with sand contains a gold nugget and repeat that six or more times so they can hit a p-value of 0.000001 or lower. Requiring a medical study to achieve that kind of p-value would cost more than the prize, because there's so many more variables in play.

Also, they'll probably simply disagree that totally changing someone's environment counts as a paranormal way to influence cancer outcomes.