Oh gawd - not again!
Quick version - the question has no answer because it contains fallacies, and any attempted answer must make one of two assumptions to remove the fallacy (thereby making it an answer to a different question). The two optional assumptions will produce two different answers ("yes" and "no" respectively).
Summary:
The aeroplane will only take off if it moves with respect to the AIR. The aeroplane really doesn't care much about how fast the ground is rolling past beneath its wheels.
The principle fallacy is the idea that if an aeroplane is propelling itself forwards at 20mph on a runway that is moving backwards at 20mph that will stop the aeroplane. It won't. There might be a miniscule reduction in speed due to increased dynamic friction in the wheels, but it wouldn't really be measureable.
So that's the core fallacy in the question as it is usually phrased* - the rolling runway would not have the effect that the question setter claims, so the whole questi0on is just bunkum. To answer the question you need to assume either:
a. The rolling runway would stop the aeroplane; or
b. it wouldn't
In case (a) the aeroplane wouldn't move forwards, so it wouldn't take off.
In case (b) it WOULD move forwards, so it WOULD take off.
But either assumption makes it the answer to a different question.
Fiunal thought - I understand there is an bill going through parliament which will legalise hunting people who post this question on forums with dogs. Or at least if there isn't there should be...
* "....on a conveyor belt runway configured to move backward as fast as the aeroplane is moving forwards" or "....on a conveyor belt runway configured to move backwards fast enough to stop the aeroplane moving forwards"