This is a typical reference frame error. The plane is not powered by it's wheels, it is powered by either a propellor or a turbine relative to the air. The thrust is placed into the air, not the conveyor. The conveyor can match the planes speed in reverse, that just means that the wheels are turning twice as fast as the plane is moving relative to the air. No matter how fast the conveyor moves, the plane still moves relative to the air. Hence, it will take off. This is a classic mis-direction brain teaser. The one caveat is that if the friction in the wheel bearings/tires create drag equal or greater than the thrust, then the plane cannot accelerate to take off. Of course that would be moot, as the wheels/tires would catch fire, and there would be a big boom fed by aviation fuel.
Enough already!! How did I let myself be dragged into this??????????
Oh, and yes, theortically thew bicycle would take off, but it would sink back down as the forward speed decreased after lift off. Now, if you put a propellor on there to keep it moving forward after lift off, then yes it would stay up as long as it is moving forward. This effect is easily explained with vortex theory and circulation. **** that Bernoulli.........................
PS, I used to be an Aereospace Engineer specializing in advanced composites...........I'm now a Automotive Machining Process Engineer.
Have a good day, I'm off to go cross eyed analyzing CMM data............................Can't wait to go home and work on boats