Quote:
The other group thinks the question means the belt moves against the plane, in an attempt to keep the center of mass of the plane stationary.


This is how I interpreted the question. In order to take off, the plane must have some sort of airflow under the wings. With no wind present, the plane must create thrust in order to achieve air flow. If the purpose of the conveyor belt is cause the plane from thrusting forward (independent of the wheels or friction or whatever)--in other words, if the backward speed of the conveyor belt is inversely proportionate to the forward thrust of the plane--I don't think the plane would take off.

Ultra-simplified example:
Airplane’s forward speed (500 mph) - conveyor belt’s inverse speed (500 mph) = the plane is essentially stationary.