You cannot give up and not even try just because you can't find any evidence. That's silly.

Check your email or facebook, see if you can find anything there. Do you have a cell phone? If you do you can request cell phone records. Maybe you can find some pictures of you taken around June 15 or some correspondence?

I'm sure there is a way to prove your presence on June 15th. If you're unsure how to do it contact a lawyer and see what he says.
You cannot not take advantage of this opportunity because you can't prove your presence or can't come up with the fee. If you feel that you qualify and have a clean record do whatever it takes to prepare your application.

You gotta take care of business.

Suffer the pain of discipline or suffer the pain of regret!

I have that quote hanging on the wall in my room and always look at it whenever I don't want to do something.