Potential downsides to requesting an extension
I suggest that if the university has set a rule that they will follow it properly. To make another assumption or guess is a fool's game.
The downside of not requesting the extension is clear; you don't do as good or complete a job. If you want to game that, then fine, but I wouldn't recommend it.
It is a mistake to guess or assume that people are trying to trick you so that they can, for some conscious or unconscious reason, punish you.

I'll note that they could have set a rule that, for example, you can have an extra week for a 5% grade penalty. But it would have been unethical to not say the last part if that was the intention.