It's relatively new (2005) so I doubt that has come up yet. I believe it's even more relaxed than the TN class visa.
I do know of an E-3 holder who had their residency fast tracked, but there have been so few of them issued that it's probably a bit hard to tell the lay of the land.
The trick is that it's a non-resident-intent visa, which is to say, while it's indefinitely renewable on paper, at some point USCIS is going to point out that you've been here for an awfully long time and you risk the renewal not going through.
Not sure if the same applies to E-3.