Which room type will you be trying to book? Are your dates flexible?
edited to add: The 2BX (2 Bd OF) is the room type which is hard to book. There are only 22 of these rooms available to book. If you are only looking to book the 2BR, it is much easier. I would expect the same availability in late July/earlier August as early June. For early June, you can currently book a 2BR and start walking. For a 2BX, I would start earlier in order to break in and walk the reservation.