Strong experimental work as usual, Iacopo! Hope to have the time (and battery) to re-digest this entire thread on the airplane today.
I too would need more time and battery for these things ! I love to think to the physics of the spinning top but it's time and energy demanding.
By the time I have become more and more convinced that rolling resistance is linked to the rising of the top, I saw this more times, in different experiments. Whenever there is an increase of the rolling resistance, there is a faster rising of the top.
In one of these experiments I used a particular kind of gyroscope, spun like a spinning top;
this gyroscope allowed me to observe the effects of an increase/decrease of the rolling resistance separately from the effects of an increase/decrease of the rotational sliding friction.
In normal spinning tops this is not possible, because the rolling resistance and the rotational sliding friction happen in the same point, the contact point of the tip, so if you change the spinning conditions of a spinning top, (spin speed, angle of tilting, kind of tip and of spinning surface), and observe the rising speed, there is no way to understand if the changes of the rising speed are due to the rolling resistance and/or the sliding friction.
The gyroscope I used allows to increase/decrease the rolling resistance leaving unchanged the magnitude of the sliding friction, so that it is possible to observe in which way the rolling resistance affects the rising speed, without being confused by the sliding friction changing together with the rolling resistance.
Then it is possible to increase/decrease the rotational sliding friction leaving unchanges the magnitude of the rolling resistance, so to see the effects of the rotational sliding friction on the rising speed.
I have made a video, (months ago), about it, which I believe it wouild be interesting.
I should put subtitles and some drawings, and post it, (when I have time).
The result of the test with the gyroscope is that an increase of the rolling resistance increases the rising speed,
or, it decreases the sinking down speed, if the gyro is sinking down and not rising.
An increase of the rotational sliding friction instead, increases the sinking down speed of the gyro, or, it decreases the rising speed, if the gyro is rising.
Breafly, I am sure that the rolling resistance is linked to the rising of the top.
What I am not sure, is whether there is another cause too, or not, helping rolling resistance to make the top to rise faster.
Because, from my tests with numbers, it seems that rolling resistance alone is a bit weak to explain the rising speed I observed in the tested top. It could be that there have been errors in the measurements/calculations, or maybe there were no important errors and in this case there should be another cause, together with rolling resistance, which helps the rising of the top.