What others have to say.....
The question is slightly unclear: if the conveyor "tracks the plane's speed" and keeps it exactly zero, then obviously, no, the plane would not take off as it would be stationary with respect to the air. I don't see what could possibly be difficult to understand about that.
The way the question is worded though, it could mean to say that the plane moves forward at 150kts, while the conveyor belt moves backwards at 150kts, for a rotational speed of 300kts for the wheels. In that case, as long as the wheels don't burn off, the plane would take off.
Perhaps the wording is all that is causing the problem there...
Looking only at friction, and assuming the aircraft could take off if unhindered the cases are as follows:
Case 1: If the treadmill is frictionless and the wheel bearings have any amount of friction, the treadmill will move (in the direction of flight), the wheels will not turn, and the plane will lift off.
Case 2: If the treadmill has any friction and the wheel bearings do not, the treadmill will not turn, the wheels will turn, and the plane will lift off as if on a conventional runway.
Case 3: If the treadmill and wheel bearings have infinite friction, the aircraft will sit there under maximum power and not move, unless maximum power overcomes the friction between the wheel and the treadmill, then all hell breaks loose.
Case 4: If the treadmill and wheel bearings both have some friction, but not enough to stop the plane from reaching takeoff velocity, either the treadmill or wheels will turn, or both the treadmill and wheels will turn, and the plane will take off (albeit with a longer takeoff roll than in Cases 1 & 2)
Case 4 is what actually happens and has a few subcases to it:
Case 4a: There is a threshold of treadmill friction where, if the wheel bearings have a much smaller friction, the wheels will turn and the treadmill will not.
Case 4b: Conversely, it's possible for the wheel bearings to have a relatively high friction compared to the treadmill where the treadmill turns and the wheels do not.
Case 4c: It is also possible for combinations of treadmill/wheel bearing friction to have both the treadmill turning AND the wheels turning during the takeoff roll.