To be offered an interview, your application needs to be sent in. Whether the scores and other school documentation gets in yet, won't hold up your app. They are assumed to follow. Once you hit "submit", you'll be eligible. Now you may or may not get offered one primarily based on the availability of interviewers in your area. I'm 99% positive that Princeton does not pre-sort applicants in order to determine whom to offer interviews.
In general, you'll be on a potential interview list regardless if you're a shoo-in or a zero-chancer. If you're an SCEA applicant, PTon definitely wants you interviewed quickly if at all possible.