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A plane is standing on a runway that can move, like a giant treadmill. When the plane's engines throttle up, it begins to move forward, but the treadmill is made to match the forward speed of the plane, only in the opposite direction. So, as the plane moves forward, [treadmill] moves backwards beneath the aircraft.
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First - what is the purpose of wheels (landing gear)? So that velocity of a plane is totally irrelevant to ground. Wheels will spin as slow or as fast as necessary so that velocity of the runway and velocity of plane stay totally independent.
Second - plane's velocity is determined by a force applied between plane and air. As engine force increases, then airplane acceleration (and therefore velocity) is according to the well known F=ma equation. Engine force determines airspeed - a relationship between plane and air.
Meanwhile, what do wheels do? See point first. They spin as fast as necessary so that plane's velocity is irrelevant to (independent of) earth.
Since air and runway remain at same location, then plane's velocity to air is same as plane's velocity relative to runway. Therefore wheels spin at plane's airspeed which is same as runway speed.
In this problem, we have added something unique. Runway is replaced by a treadmill. As plane moves forward 1 MPH (relative to air as in point second) then treadmill moves 1 MPH in reverse. Again, wheel's job is to keep airplane and runway completely separate (point first). Therefore wheels must now spin at 2 MPH to keep treadmill and plane independent of each other.
If wheels applied the "F=ma" between plane and runway (as in a car or bicycle), then wheels and ground would not be independent. But wheels do not move this plane. Motion is created completely by a relationship between air and plane - its jet engine. No matter how fast ground moves, plane's velocity (and acceleration) is only based upon F=ma between plane and air. No matter how fast ground moves, wheels will spin as necessary to keep treadmill and plane completely independent of each other.
Now we address other possible (and unstated) references.
Since treadmill will always move backwards (relative to air) at the same speed that plane moves forward (relative to air), then wheels will spin at twice the airspeed.
Treadmill is set to match forward airspeed. Relative to what? If treadmill's speed is relative to air and plane's speed is relative to air, then wheels spin at twice airspeed. But if treadmill's speed is relative to plane, then treadmill never moves relative to air. Now we have wheels only moving at one times airspeed.
One fact we do know. Airplane's speed is always relative to air because those engines create a relationship only between air and the plane. Treadmill's speed (in this problem) could be relative to air or relative to plane. But again, point first - those wheels always make ground independent of the plane. Wheels will always spin as slow or fast as necessary so that plane and earth velocities do not affect each other.