Challenging claims is a process best done with delicacy and foresight. In particular, it's best to pretend to be humble, collect lots of data, show your work/thinking in detail, and understand the psychology of your adversary.

Be prepared for disappointment; be prepared for vicious attacks. Expect that the power structures of the establishment will make things uncomfortable to you. But continue on, humbly collecting your data and showing your thinking.

I've watched several key paradigms fall after 20-30 years of dogged work.