We'll be there from the 5th to the 12th!!
Here's my reasoning as to why I think the crowds will be smaller after the 4th:
It's soon after schools close, and many people do their summer vacation mid-summmer than right after schools close.
People are expecting it to be crowded for July 4th and they think the crowds will linger after the 4th, so they go early in July/late June to avoid the 4th of July left-over vacationers.
I can't think of any more reasonable explanations, but count me in for hoping it's not massive.