If the judge sets out for him to prove he eligble for office, he’s going to have to come up with some documentation that proves it. Saying the dog ate my homework probably won’t work on a Marine.
I suspect what will happen, before any judge at any level would order defendant to ‘show proof’, is discovery, to allow both sides to collect and enter exhibts for use in the case placed under jurisdiction of the court. If the discovery phase is extensive enough to track the mans’ citizenship and any use of foreign citizenship if it occurred, that would be sufficient to most likely cause the lying sonofabitch to resign and step down, THEN he can be prosecuted for massive fraud.