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Unless another spell or ability is affecting your opponent's maximum hand size, Jin-Gitaxias's ability will result in your opponent having a maximum hand size of zero. They will discard each card from their hand during their cleanup step.
If a player has more cards in their hand than their maximum hand size during the cleanup step of that player's turn, that player discards until they have that many cards. A player's maximum hand size isn't checked at any time other than their own cleanup step.
Jin-Gitaxias doesn't affect the maximum hand size of its controller. Because the cleanup step is after the end step, its controller may have to discard some of the cards that were just drawn.
