"subject to jurisdiction" does not mean "has to follow US law", that is territorial jurisdiction. Subject to jurisdiction means political jurisdiction and allegiance. That is why children born to members of Native American tribes (no matter the location of birth) were considered not subject to jurisdiction because the parents held allegiance to their tribe. Read United States v. Wong Kim Ark (1898).

I don't have an opinion on the legally correct answer, reading the full decision and dissents I'd give a slight edge to the majority.