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Any persistent card that triggers "at the end of your next X turns" is intended to trigger at the end of your current turn as well. I know it's not entirely consistent with other terminology, but there is no succinct way to say "on your next X turns, including this one."
why bother with 'next X' at all? it's implicit because you've got a progress meter on the card. so you probably would have been better off saying "at the (beginning/end) of your turn, do (thing)" and rely on the progress tracker to communicate how long it will last...