"I think the descent speed of the airplane is not that much"
I think you are wrong. Airplanes can descend very rapidly when needed.
The airplane is aerodynamic, meaning it slices through the air nicely, whether it is flying straight-and-level, or aiming straight for the ground. So its free-fall speed is much higher than that of a roll of paper, which will have considerable wind resistance.
In addition, the airplane has a propeller, which adds force. When in a dive, the airplane has gravity plus the force of the propeller taking down. But the roll of paper only has gravity.
I expect that in a full dive, the airplane could probably exceed a 10,000 FPM descent, or around 120MPH, which is around the terminal velocity of a skydiver.
(pulling out of a full dive safely is not easy, however).
The roll of paper will obviously and intuitively be slower than a skydiver.
I estimate it to be no more than 2,000 FPM, or about 20 MPH. (and probably a bit less)
(edit: In the comments, Henning Makholm suggests that a paper roll may only fall at 200 FPM. That is less than 3 MPH, which seems a bit slow to me. A slow descent rate for a Cessna when approaching an airport is 500 FPM.)
The airplane does not even need to enter a full dive to be faster than the paper. It could probably catch the paper with a nice, gentle 2,500 or 3,000 FPM descent.