I have carefully noted the posts with bogus ideas so I can qualify future advice given by those authors.
Listen to everyone but qualify your sources.
I didn't see what the dificulty is. The original post clearly stated the givens and the outcome should be obvious to the most casual observer.
Since the conveyer belt runs as fast as is required to keep the plane standing still with respect to "the world" there is no wind over the wings except as provided by local weather and I suggest there isn't enough wind to equal or exceed the speed needed to let the craft lift off. Actually, unless the person stating the problem allows for wind, there isn't any.
The air is standing still with respect to the world, i.e. no wind. The aircraft is standing still with respect to the world due to the clever control of the conveyor belt. So the aircraft is standing still with respect to the air. No air flow over trhe wings eguals no lift.
If we temporarily lift restrictions of maximum thrust of the aircraft and make it super powerful, eventually, at enourmous speeds (hypersonic), the drag of the surface of the conveyor belt will propel enough air down the "runway" to provide enough lift such that the aircraft will lift of the runway with no apparent forward motion with respect to the world.
Once free of the conveyor belt and airborn the aircraft will be propelled forward but if the conveyor belt/runway does not extend forward of the aircraft a sufficient distance or if the conveyor belt is slowed to a stop the plane will return to the ground (landing or crash? can't say, not enough information)
If the aircraft doesn't develop enough forward speed with respect to the world and its zero wind before leaving the region of conveyor developed wind the pland will lose lift and fall out of the air. Not enough information is given to know if the aircraft will developp flying speed before abruptly leaving the region of conveyor accelerated air.
Ignoring the super powerful propulsion idea for the plane and a hyperfast conveyor belt, the problem is quite trivial. Aircraft like sailboats are governed by relative wind. The only relative wind (above zero knots) in the original problem statement is that blown over the wings by the prop if there is one. Jets would have much less. Developing enough lift to take off using the prop blast is possible but not achievable to the vast majority of aircraft.
If we can use an F-35 or a Harrier jump jet or an Osprey then the problem as stated is meaningless so it must be assumed these special aircraft were not to be used.
A practical concern would be overheating a piston engine. You would be at max throttle for an extended period and only have the limited airflow created by the prop with no cooling assist from "ram air" due to motion through the air.
Oh by the way, the pitostatic system would indicate zero or near zero airspeed and that is what counts for wings generating lift.
Pat