Deducing strict parts

Consider an object oriented ontology. Let A and B be objects and let A be a strict part of B. Now, there must be a truthmaker for the truth of the proposition "A is a strict part of B". This propostion is logically equivalent with "A is a part of B and B is not a part of A". It's truthmaker is not confined to A and B. For A and B together only make it true that A is a part of B, whereas also proposition "B is not a part of A" needs to be made true. So which object or objects make "B is not a part of A" true? If B is not a part of A, then there must be an object that is part of B and that is not part of A. So a truthmaker of "B is not a part of A" is any object that is part of B and not part of A. It follows that B has at least two strict parts, namely A and the aforementioned truthmaker of "B is not a part of A".

Do truths always have ontological import?

Could there be truths without truth-makers, without truth-bearers, and without any other kind of ontological footprint? In other words, could there be truths in a possible world that lacks concrete objects and that also lacks abstract objects such as universals and propositions?