Maybe not. Is the princess subject to the jurisdiction of the United States? I would assume that she would enjoy diplomatic immunity and therefore would not. So under the 14th Amendment that may prevent automatic citizenship. An interesting legal dilemma.
A parent has to file a consular report of birth abroad (CRBA) and the child’s citizenship at birth is filed. That action by the parents would indicate jurisdiction of the United States.
I understand that if the parents are negligent, then the child can be issued citizenship papers in acknowledgement of his/her citizenship at birth. That again would be an acknowledgement of US jurisdiction.