The Pledge
"I pledge allegiance to the Flag of the United States of America, and to the Republic for which it stands, one nation under God, indivisible, with liberty and justice for all."
You will note its a pledge of allegiance to the flag and the republic - NOT to the President - therefore I'd say unless he is able to change the constitution his office is subordinate to the state and the constitution so running for a 3rd term is to stage a coup.
In any event he will be almost 83 at that time - 5 years older than Biden was when he was elected as the oldest Pres. He will be lucky to be alive at that stage and in any event he is surrounded by lick spittles and traditionally when a leader gets old and infirm at one or more of them will fancy their chances and turn back stabber