Airplanes take off because the lift on the wings exceed the force of gravity on the airplane. Ground School 101.
Lift is created by air passing over the wing. Which is effected by the shape of the wing (think about flaps), angle of attack (Think rotate to climb pitch attitude) and speed of the wing through the air (Think Vr, Vx, Vy).
The only other effect at low altitude is Ground effect which is based on Wing generating lift and proximity to the ground.
The conveyor belt is nothing more than a a land based version of a float plane on river and how the current effects it's take off.
1) Can I setup the conditions so that an airplane on a conveyor will take off? Yes
2) So the answer to the question in the post is Yes? No, I can also setup the conditions so an airplane on a conveyor belt cannot take off.
3) So the answer to the question in the post is No? No, see question 1.
Replace the word conveyor with runway there is still no answer. Think Density altitude.
This is making my head hurt