After a brief search I can’t find any mention of any punishment for Sisyphus if he stopped pushing the boulder up the hill. That said, this is all taking place in Tartarus so there probably isn’t much else to do and endlessly push a boulder up hill might be the better option lol. I can relate.
My understanding is that all of the codecs we are discussing are deterministic. If you have evidence to the contrary I’d love to see it.