the Federal courts have handed down a number of rulings which have resulted in it becoming very *difficult* to lose one's US citizenship.Unless he renounced his citizenship *before a US State Department official* that renunciation wouldn't be valid...wouldn't be recognized.An important case is Action and Deltamar v. Rich, 951 F.2d 504. There the court ruled that Marc Rich (yes that Marc Rich) retained his American citizenship despite specifically renouncing it in a Spanish naturalization oath.
The court ruled that Rich's subsequent actions, including using his US passport, showed that he had no intention of giving up his American citizenship.
Rich wasn’t running for President.