The question you need to answer is: does the treadmill increasing speed increase the force pushing the plane backward / stopping it from moving relative to the air?
If the answer is no, then the plane takes off. If the answer is yes, then the plane may or may not take off, depending on the maximum force the plane can push itself forward with using the engines compared to the force the belt can apply backward.
First, to clarify: Imagine this problem, but in a wind tunnel. In a strong enough wind tunnel, the plane will simply lift off the belt. It need not move relative to the belt to take off, it only needs to move relative to the air.
Second, the behavior of the bearings. If the bearings are "idealized", then it doesn't matter how fast the belt is moving, it only causes the wheels to spin, but doesn't push the plane backwards. In this case, the plane would obviously take off, the wheels increasing to very fast speeds, until enough lift is generated that they stop contacting the belt and the plane has taken off. If the bearings are "anti-idealized", ie there are no bearings, the "wheels" are just glued to the plane, then it depends on the effective coefficient of friction between the wheels and the belt. If the wheels were PTFE on a PTFE belt, for instance, the plane would likely still be able to take off. If it was high friction (ie normal) wheels, on a high friction belt, with no bearings, then the plane would not be able to take off, since the force applied by the belt would overcome the engines.
Finally, the real case. The bearings on the plane's wheels are not "ideal", they transfer a nonzero amount of force to the plane, but it should be very small. I haven't "run the numbers", but I'd be really surprised if the plane wouldn't be able to take off. Imagine how the wheels interact when landing and taking off normally: the moment the wheels lift off, you don't feel a lurch forward as the force pushing backwards disappears, and when you touch down (continued)
