tbh, I'm betting the beta or Alpha phase 2 will be starting begining of June.
SE said themselves that alpha 1 will be capped at 20 (and a lot of people are at that level in the alpha), and that they wanted to start alpha 2/beta 1 as soon as possible, but they wouldn't do it until the servers became stable enough.
Well, the past few tests have been very smooth, and this is the first week 3 tests in the same week have taken place. So if next week goes just as smooth, I believe it is rather safe to say alpha 2 or beta 1 announcement will take place beginning of June, and likely not start later than middle of June.
Now, it is possible they hold out for E3 and have a "beta invite" event at E3, but I find it more likely they'll make some announcement at E3 with a new trailer, and a more in-depth explanation of the way they wish FFXIV to take, as well as an announcement at some new classes, beastmen and storyline etc. They might also offer a playable version of the game.
The game stop etc. date is only a preliminary date. If you seriously believe SE planned the alpha to go smoothless and state that FFXIV will be delayed because of how sloppy the beginning of the alpha was, you're rather stupid imo. No company believe an alpha version of the game will run good. Alpha and beta is just that - an alpha and beta version of the game, which is offered to people with the some intent of finding bugs, errors etc. It isn't meant to be a fully functional playable version of a game.
Look at how sloppy the alpha was in the beginning,and compare it to how smooth it is now, 2 months after SE released it. That if anything is a clear indication that they're on track.
And no, I don't care how much you argue how smooth the FFXIV NA beta was. That wasn't a beta, it was a localization. The game has already been out in Japan for months.