To better visualise my fictional attractive force, I will give three examples how it would work in real world.
1. If an asteroid is so called pile of rubble type, when it rotates too fast, centrifugal force would cause it to fly apart. But there are some asteroids which defy standard expectations, they rotate too fast for standard gravity to keep it together, yet they dont fly apart. If my fictional attractive force would be real, one of its effects would be exactly the same, holding fast rotating objects together thanks to counterbalancing centrifugal effect. But it is important to mention, that this doest not mean, that the attraction force helps the asteroid to spin up, it just attracts the parts ot the asteroid toward some point under surface (but not to center for most parts). On the other hand, scientists have come up with other solution, they think it is held together by van der Waals force. But nobody really confirmed this hypothesis by some experiment or some detailed study. One example of this effect is asteroid 1950 DA:
https://www.space.com/26819-potentially-dangerous-asteroid-1950da-rubble-pile.html
2. If this fictional attractive force would help to sustain rings around planets or asteroids, it would mean that if the rotation speed decreases over time, the existing rings would just drop on equator and then it would then look like this:
https://en.wikipedia.org/wiki/Equatorial_ridge_on_Iapetus
3. If there is some big and fast rotating planet, at some point in the orbit of the planet, which is near to range of maximum of this attraction force, there should be visible some rotating spirals, which would show combination of gravity and this fictional attractive force. It would look similar to this (already posted in one of previous replies):