Honestly we might not find out for sure even if legal action is taken - if he does end up in court, it will still be 100% in his interest to tell them he's trans even if he's just a teacher who got sick of being told he couldn't say anything anti-woke and decided to make his employers live by their own insane rules.

The tranny ideology of Self-ID is completely unfalsifiable, so it's not even a lie in a legal sense.