2014.
They would, and there are agreements about that. The problem gets to be that with the lack of status in the US, you wouldn't be in a good position.
On the other hand, it is a very, very good reason to come three months early. It doesn't undermine your case in the same way that someone coming just for fun would.
It's a gamble. Nobody can tell you what the agent will do. The agent can let you in, he can turn you around and you then have to leave.
If it were me in your position, I'd try it before the end of my US status, and be prepared to buy an expensive plane ticket if I get turned around. I wouldn't be too worried about the denial, as it doesn't make you look bad in the same way that smuggling or violating the law does. That's based on my personal risk profile, though. I'm a gambler who deals with risk by having plan A through plan J.
The safest option would be to fly home and return, but that comes with expense. Only you can decide if the risk of rejection and needing to explain on future applications is worth the risk of having to lease a place for a few extra months while you fly back to your country of nationality.
Two years is fine.