If your arm is hanging outside the window and the vehicle is actively rolling, you're likely going to lose that arm.

Otherwise, based on all the other logic described, the decision to roll the windows up probably takes into account if there's an obstruction (just like auto-up windows do).

It will help keep your head from bobbing outside the vehicle as it rolls, preventing your death from a crushed skull.

Edited to add:

"partial ejection" are the magic google words. Here's an example:

"It was found that, in standard impact tests on high containment barriers, partial ejection of the head through the side windows occurs systematically"

from: https://trid.trb.org/view.aspx?id=750825

If you're in a rollover you won't have great control over your arms in the first place, so if your windows are open there's a good chance they'll be going out the window at some point. If you watch professional drivers in accidents, they attempt to cross their arms across their chest to keep them from flopping around.

I'm not sure of the official reason, but I'd imagine it prevents ejections (or worse, partial ejections) from the vehicle.

I was the first person on the scene of a rollover where a woman was pinned underneath the car (fortunately only suffering broken bones) because she was partially ejected through the open window.

As someone who frequently drives with my arm out the window, that would certainly be concerning. I can see some considerable safety benefits to containing unrestrained people in the vehicle (rollovers tend to result in ejections, which have a high mortality rate), but at least from a management perspective, I'd much rather let a negligent person be ejected 50 times than keep 49 of them in the vehicle and have 1 sue us because we cut their arm off.

After extensive testing it was found that a clean amputation provides a quicker recovery than a degloved, torn up limb. There were more amputations that would be ideal, but net morbidity was reduced.

If it prevents a child from being ejected, it will likely save their life.

If your arm gets flung out the window when that side of the car is facing up then it could get crushed when that side rolls over.

"Keep your arms and legs inside the vehicle at all times."