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According to the Montreux convention, Turkey should block the Bosphore to any military ship, unless they are returning to their home base.
In time of war, Turkey not being belligerent, warships of belligerent Powers shall not pass, except i.a. to return to their base. (art. 19)
I'm not sure how this military ship can enter thr Black sea.
Looking at your quote they should be allowed passage since the US is not formally a combatant (i.e. belligerent) in the conflict in question.
they will not enter the Black Sea, they will put high pressure from the entrance, like a golgoth outside a room that you know will transform you into a moskva if you dare exit the room. even if you stay in the room, you still fill that pressure ^^