I'm interested in knowing when primaries will first be sent to medical schools, too! Also, OP, I think applications could have been submitted before June 2. As far as I know, they could have been submitted the first day the application opened in May.
A related question: how long does it take to get verified once your application is complete?
Another related question: how long does it for AACOMAS to mark transcripts as "received"? My school sent my transcript on June 3 by first class mail, so it should be delivered to AACOMAS by early next week. How long should I expect for it to take before my transcript is marked as received?
(Sorry for hi-jacking your thread, OP!)