An open presupposed proposition is a presupposition that contains a variable about which the utterance supplies information.
This proposition is presupposed by the contrastive stress in she gave the SHIRT to Harry and by the cleft sentence it was the SHIRT that she gave to Harry.
Prince, Ellen F. 1986. "On the syntactic marking of presupposed open propositions." In Farley, Farley, and McCullough 1986