Would Truman's immunity from the 22nd amendment in theory allow him to run for unlimited terms?

As the current president at the time the amendment was passed, Truman was granted immunity from the 22nd amendment, allowing him to run for a third term despite serving more than half of FDR's.

He chose not to run for a third term given his low approval ratings, but say he did and won. Would that have been the end of the road for him or would he have been able to run ad infinitum like his predecessor? Imagine a world where Truman is president until December of '72!