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... or possible make it so the "total game" timer does not start to run until 24 hours past the last move - so the first 24 hours from the last move is a "free" time that would not hurt anyone.
I still don't see how playing one minute after someone is going to actually hurt them.
Under your scenario, both players would at least be making one move per day.
I don't think anyone wants a tournament where you can get timed out while making a move every day.