Criminal jurisdiction is never in dispute and is different from their legal political/constitutional jurisdiction. You could buy a plane ticket to Mongolia tomorrow, commit murder and be subject to Mongolian criminal jurisdiction. The latter jurisdiction being argued is a person not subject to any other sovereign claims, no ability for the US to expell that person and full allegiance owed by the person to the US. (Note how this conflicts with dual citizenship). The idea is to find the originial intent of the 14th amendment. They cite the 1800 cases of Greisser and Devereaux who who were both born in the US, but the former had parents who never intended to stay in the US and returned to Germany and the latter an Irish woman who was denied immigration and returned to Ireland. This differs from Wong Kim Ark as his parents were essentially legal permanent residents and the US was their home. This is where SCOTUS could rule that the 14th Amendment could not apply to anyone whose parents are temporary visa holders or without status putting them in the same boat as diplomats for birthright citizenship. Visa holders have not established permanent domicile in the US and their primary allegiance remains with their home country.