Let’s say the airplane needs 100mph ground speed to take off.
Let’s also assume that the landing gear can support the plane at 200 mph (forward speed of the plane, plus the backwards speed of the conveyor).
If the engines can overcome the additional forces against the landing gear of 200mph over 100 mph (not a big deal) and the landing gear can withstand twice the limit of ground speed (again, not a big deal) The plane will take off.
The question states that the conveyor matches the speed of the plane. The plane moves 100mph forward (ground speed) The conveyor moves at 100mph backwards (ground speed). The wheels turn as if the plane was traveling 200mph.
What’s to understand? The plane is moving 100mph down a moving runway moving in the opposite direction. So what if the runway is moving? The plane has 100mph air over the wings. It takes off.
To put it another way  What is the frictional force against the airplanes wheel at 100mph?
What is it a 200mph? Three times the resistance? Four?
Now, we also have to deal with the resistance created by the rotational velocity of the tire spinning up to 200mph, instead of 100mph. Race cars overcome that routinely.
Consider the wind resistance as a plane approaches take off speed. It must be 100 times the resistance from the landing gear. The landing gear resistance is nothing.
It’s gonna be there, for sure, but hardly something that couldn’t be overcome.
How about this? Would it be possible for a plane to take off with a 100mph tail wind? Same thing. Not something anyone would dare to do, but something that would be about as easy as building the hypothetical treadmill.
