I've always pushed my Aeropress to the end. The "whoosh" sound is something to be relished.
Honestly I don't think it would make much difference, but more of a 'belief'. Unlike espresso, the pushing on Aeropress will not add any bitterness/overextraction, it's more of an act of filtering the coffee. But if you press the coffee bed with the plunger, that *might* have some effect, but not significant in my experience(could be due to my newbie palate