I got bored of trying to read this after the 4th page...
if this is a typical aircraft like a Cessna 172, and the conveyor belt is a thousand or so feet long, with no tailwinds, and the brakes are released, and they don't turn on the conveyor until after the engine runs up to power, and the plane is not tethered to something... my answer is YES, the plane will take off.
With the brakes released, they will spin whatever speed the conveyor belt is moving, plus or minus the ground speed of the aircraft. If the conveyor belt is spinning at 20mph going "south", and the plane is not moving (tethered), then the tires will have a 20mph speed. If the plane's engine+prop produce enough thrust to move the plane "north" at 20mph, then the tires will be have a 40mph speed. And if the engine/prop moves the plane 20mph "south", the tires will have 0mph speed. And, if the engine/prop move the plane at 100mph north, the tires will be at 120mph for a short period, then slow to 0mph unless pilot is stupid and keeps pushing the yoke in.
To bet yes or no, I'd have to know all of the stupid little specifications of what mythbusters are gonna do. If they don't mess up too much, the plane will take off.