Why aren't there screw-threads inside the Aeropress sleeve?

I'm staying in a hotel with nothing but paper cups in the room, and I'm not travelling with my usual suitcase in which I stash my emergency polypropelene folding cup, so I'm reduced to making my hotel coffee using the awkward hold-the-sleeve method, in which you grip the sleeve as hard as you can with your left hand while pushing down on the piston with your right, supporting the press so you don't crush the paper cup beneath.

This got me thinking: why isn't there a screw-threat inside the Aeropress sleeve, matched to a thread on the piston? Instead of pushing down on the piston, you'd just give the thread a quarter-turn every few seconds as the water pushes its way through the coffee, meaning much less physical force and no worries about crushable cups.

(Allow me to forestall the upside Aeropressers here by noting that you are bananas, seriously. Just do a blind taste-test and if you can distinguish the difference, I will acknowledge that your tasting abilities are far superior to mine.)

I think that this would confer a huge mechanical advantage to the press-user, as the screw would give you the leverage to force the piston down, rather than relying on your muscles. On the other hand, I imagine the threat would get coffee fines caught in it and be harder to wash out. But there must be a way to finesse this?

(Image: Netanimations)