if it was that player's last territory you will lose. say you had all the capitals but one, green for instance is not your target but someone else's. green has the last capital you need and its the only territory he has. you take that capital and green is gone. his assasin wins the game. you must hold the capitals at the START of your next turn. not at the end. that next turn will never come, because the game is over. when one player is eliminated, an assasin game is over right then.
if no player has been eliminated, and you hold the capitals (or whatever victory conditions on one of the several maps that have objectives) then you would win the game when your next turn started, and the assasin thing would be meaningless. this would also be the case should the assasin game be played with a round limit and the limit came and you had the most troops.