Ah, but you moved. That totally eliminates the original question. Now, if you are the observer, then I don't move forward relative to you, since you and I are moving forward at the same speed. Relative to the sidewalk, I am moving forward faster than it is moving backward and will continue to do so as long as you are pulling me forward, but my position relative to you will not change. Thus, to you, I will appear to not be moving at all.
Now, if you modify the original question and throw in the detail that the aircraft moves forward relative to the ground observer, then yes, the aircraft will lift off once it has achieved enough forward airspeed. After all, it is all about airspeed. Ground speed is irrelevant. Right now, we are all moving at about 1300 ft/sec relative to the sun, but since the air around us is moving at the same speed, on a calm day, we have zero air speed.

For clarity, this whole thing is dependent upon relativity. Assumptions are that the airplane is moving 300mpg forward relative to the conveyor. The conveyor is moving 300mph backward relative to the ground observer. Therefore the aircraft is moving at 0 mph relative to the ground observer. Now, if the plane is moving forward 300mph relative to the ground observer, who is stationary relative to the Earth and the surrounding air, then yes, the plane will take off, assuming that it takes off at 300mph. but, if the plan is moving forward at 300 mph relative to the ground observer and the treadmill is moving backward 300mph relative to the ground observer, then the plan is moving forward 600mph relative to the runway (conveyor belt). So in reality, it's a flawed question and your answer depends on your point of view, literally.