I thought that was a valid point myself but no. They’re not subject to the jurisdiction of the US.
Wrt to “subject to the jurisdiction”
I’m not a lawyer but have heard of the concept “under the color of law” used in association with determining jurisdiction of illegals, by using the claim that any governmental agency recognizing the illegal (for instance, by giving welfare benefits or a driver’s license) gives them “color of law” and therefore legitimizes their presence ex post facto.
Needless to say I find this complete BS and very dangerous.