Can a mentally incompetent defendant challenge his conviction under the Fourteenth Amendment despite procedural rules?

QUESTION PRESENTED ~The Due Process Clause of the Fourteenth Amendment prohibits states from trying or convicting a defendant who is mentally incompetent. See Pate v. Robinson, 383 U.S. 375 (1966). Under the Fourteenth Amendment to the United States Constitution can a man who has been convicted when he is incompetent challenge his conviction notwithstanding procedural rules in the State and Federal Courts?