In these cases usually it depends how quickly they send the physical box.

There maybe a delay when you can start on June 23rd depending when shipping occurs.

Rule of thumb with the last CE from ARR is the early access code give you access from the time early access begins still the day of release.

Depending when severs restart on day of release you will not be able to log into the game until you key in the CD-Key or code from the box itself.

I might be wrong, they decide to do something different but I would expect it would happen as I explained. The early access code will tie you between early access up to the date of release.

There may or may not be downtime between early access and the game going live but the box should contain a code for you to continue after early access.