Look, let's be fair, I'm all for testing a theory.
Take cheek swabs of, say, 100 people. A mix of men and women, including those who identify as trans, and including some who have undergone surgery/take hormones etc. etc. Any proportions you like - no need to be 50/50 men vs women or anything predictable like that.
Keep a record of which swabs come from which individuals, their sex as recorded on their original birth certificate and their gender identity as reported in each case. Assign each person a number and write it on the corresponding swab.
Give the numbered swabs to a lab to test 'blind'. Don't tell them how many of the 100 fall into each category and certainly don't tell them which swabs relate to each person. Ask the lab to report back the sex of each sample - writing it against the sample number on each case. They may also flag any samples they feel are ambiguous and could do with further testing.
Put the results into a table and add back in the columns of sex on original birth certificate and reported gender identity for each swab.
Every person making the claim the OP sets out in the OP - and every person refuting it - can then be invited to guess how many of the test results will match sex at birth vs how many will match gender identity. They can also be invited to bet a sum of money against their guess.
Like a fun sort of 'guess how many sweets in the jar' for Easter!
All proceeds to be split between correct guessers to donate to the charity of their choice.
Yes?
I wonder what the results will be!