"Eustace Bright" asked me if this has ever been tested. I'm sorry to say I never gave a very good picture of what would happen if you tried to test it and why. I tend to favor theory over practice, but theory divorced from practice quickly becomes aimless rambling. So, with that in mind, let me try to start from a real-world perspective this time.
The short answer is that some aspects of the problem would be extremely hard to test. Now for the full answer...
The Wings and Engines
I said before that with "powerful enough" engines, the plane would take off regardless of how the engines themselves work. I understated the point a little bit that for some types of engine "powerful enough" might mean "like the warp drive from the Enterprise". With big propellers and a very light plane, there might be a chance it could take off from a stationary position. I really have no idea how powerful of a rocket or jet engine it would take to create enough airflow, if it's even possible. Also keep in mind that more powerful engines tend to be bigger and heavier. If the engines are attached to the plane, they're going to have to meet a certain power-to-weight ratio.
In other words, it may be possible to build a plane that could lift off in place (with only horizontal thrust), but if you pick a plane at random, you could be reasonably certain that it wouldn't pass the test.
Remember that a quirk in the problem description requires the plane to be stationary without necessarily providing the forces to hold the plane stationary. This creates a weird kind of reverse causality where you can try to find the answer and find that you've lost the question. The upshot is that if you can show that no such force can hold the plane stationary, you thereby prove that the problem has no exact solution.
I had listed two candidate forces to hold the plane stationary, friction and rotational inertia (the resistance to spinning faster or slower) in the wheels. The friction I'm referring to is in the wheel bearings. As far as rotational inertia, I think I need to clarify what I meant just a bit with another quick example:
This time, imagine a ball on a treadmill. When you turn the treadmill on, the ball moves back a bit before it gets rolling. Once the ball's spin gets up to speed, its momentum alone keeps it spinning, like a ball spinning in mid-air except supported by the treadmill. When you stop the treadmill, the ball will "coast" forwards quite a ways before friction finally brings it to a stop.
I had suggested that those two forces would be enough to keep the plane stationary, but then I discovered a flaw in my reasoning: neither of those forces is proportional to the speed of the treadmill.
The sliding friction in the wheels is a constant force, regardless of speed. Unless the wheels have some serious wobble or mechanical problem that makes them start to lock up at higher speeds, we'll only ever get a fixed-strength force from the friction. If the engines exceed that force (which is extremely likely), there's nothing we can do with the conveyor belt to compensate.
The rotational inertia is a little more promising, but not much: its associated force won't be proportional to the speed of the belt, but it will be proportional to the acceleration of the belt. Momentum and inertia are related to velocity whereas force is related to acceleration. Remember that once my hypothetical ball was up to speed, it stopped moving backwards without any additional forces having to act on it (not even friction). That means with the conveyor belt spinning at a constant speed, the wheels will offer no real resistance to a constant force from the engines.
My conclusion on the "exact match" constraint between the belt and wheels is that it very well may be impossible to balance the forces and create a stationary plane to meet the requirements in any practical scenario.
Postscript: An Apology
Last time I mentioned Mythbusters and some YouTube videos that claimed to solve the airplane/treadmill problem, but didn't seem to have anything to do with the real issues. They demonstrated exactly that the plane wasn't stationary, so they couldn't have been answering the question exactly, which had specified that the plane must be stationary. Well, that was true, but they were trying to answer the follow-up question of whether any such real-world forces could hold the plane back against the force of the engines, which turns out to be a tough part of the problem. Since the friction is a constant force, they've definitively ruled out friction as a candidate in any realistic scenario.
That said, I still have some ideas to kick around before I'll concede that the problem can't be saved. It'll be a longshot, though.
Edit: I did some more thinking and ruled some ideas out. I think I'll leave the whole airplane/treadmill thing alone for a while. It's more fun to discuss it in person, anyway.