if i were you,
i'd seek-out an Immigration Attorney (or some-such person) for guidance & advice & direction & assistance.

i wouldn't be getting info in this forum.

from time to time,
i listen to Power106FM radio from Jamaica, where they'd have an immigration expert answering phone calls.

There too-too many what-if's and other fine-print thing-ys associated with this topic.

additionally,
you may want to contact all-three embassies (i.e., US, Italy, Jamaica) for guidance.

go to the experts.