Why wait?
Who cares when it's due?
If you're on the critical path, if someone is waiting for your contribution, ship now.
We have deadlines for a reason, but the key word is 'dead'. In fact, you don't have to wait for the deadline or get anywhere near it, especially if you want to speed things up.
Too often, we find ourselves using the deadline as the lever to overcome our fear. If you're relying on drop dead dates to push yourself, the project is paying a price.
The bias is to slow down because otherwise the boss will just give you more work to do. Are you still stuck in the us/them dichotomy of factory work?
All other things being equal, faster wins.
PS the challenge with being an initiator of projects is that you are never, ever done.