The plane cannot move forward relative to both at the same speed. Not going to happen, unless the fixed observer is also moving forward in space, in which case he wouldn't be fixed. A touch and go has nothing to do with it, unless it's a Harrier dropping straight down and going back up, which is outside the scope of the question. In the question the plane is not moving forward in space. In order for it to match the rearward speed of the treadmill, it can't move forward or backward of it's present position in space. Basically, you are trying to tell me that the treadmill is spinning the airplanes wheels and thus it takes off. While there is some airflow from the treadmill, there is not enough to generate lift. In order for lift to occur, the plane has to move forward of its present location in space. Since is is only moving forward of its current position on the belt, it is static in space. The engine just keeps it moving forward on the belt at the current speed of the belt. Now, if you applied more thrust and moved it forward along the belt faster than the belt is moving backward along its track, then yes, the bird will fly, but then it has exceeded the parameters of the question. If you really think the conveyor will just spin the wheels, you're not being realistic. Sooner or later the force of the belt pushing backward will overcome the "at rest" inertia of the plane. It will need forward thrust to maintain a forward speed equal to the belt's reverse speed, i.e. maintain its place in space.

I think this is a question dreamed up by philosophy majors as payback to engineers for making the following joke:
Q: What's an engineer say? A: How Does it work? Q: What's a scientist say? A: Why does it work? Q: What's a liberal arts major say? A: Do you want fries with that?