If your game demo is complete, but you're wondering - "Should I submit now, or should I wait until the end, just in case I find a massive bug and need to do an update??" - here is my advice - SUBMIT NOW.
In order to be fair to all contestants, I download all the content at the close of the submission period. So if you submit now, you simply would need to update the file(s) sitting at your download link in order for me to get the most up to date build on June 3 when I start downloading all the submissions.
If your demo runs, you're better submitting now just in the small case something happens and you forget. At least what you have will be in. Don't get left out!