In a word, yes. Re-opening the case would at least make it possible to identify incompetent defense, confessions by frightened intimidated or learning-challenged people, and so forth.
Lots of polygraph-cheating involves taking various drugs like beta blockers, and that's why I mentioned blood tests. Psychopaths who are emotionally flat are identifiable by a competent polygraph test, so you just re-open, investigate, and re-instate.
And I'm perfectly willing to include various "truth drugs" to the protocol, given the stakes here and the apparent forfeiture of lots of privacy rights by the death row inmates.
I'd rather leave a couple of these people in jail for the rest of their lives than execute innocent, frightened, confused people.