There's plenty of dispute about what "subject to the jurisdiction thereof" exactly means. Does it mean that arresting authorities in the United States who arrest illegals for crimes committed here DO NOT have to notify the Mexican consulate or embassy of their arrest by virtue of their being "subject to the jurisdiction" of the United States and therefore not subject to any Mexican jurisdiction? Teh government of Mexico is claiming that they must be notified as the arrestees are still Mexican citizens and subject to Mexican jurisdiction. The case of Jose Ernesto Medellin is on the Supreme Court's docket for this term. We'll see.
There is nothing secretive or complex about jurisdiction when the issue is "in personam" jurisdiction for purposes of constitutional application. If a living human being is within the borders of the US, including any territory or possession, that person is subject to the jurisdiction of the American law and the judiciary for all purposes. The attempt to create some fatuous distinction and non-extant ambiguity is meaningless. The question is so finally resolved that it wouldn't even make a good constitutional law I exam question.