I can tell you that last year the first part of June was very busy, but that had a little to do with the debut of 2 new rides (little Mermaid and Star Tours) and a new parade (Soundsational.) I believe they opened June 3rd, but were soft opening before that.
Keep in mind that even though Carsland will not officially open until the 15th, there will be cast member previews, AP previews, and probable soft openings before the 15, that could also bring in people to the parks.
My guess is that it will be crowded on non local blackout days as everyone tries to get in their last trips before the summer blackout hits and also hope they soft open Carsland while they are there. Do a little planning and use fast pass efficiently and you'll be fine.