It's a lubrication problem, they don't put enough lubricant into the optical tube mechanism and it wears off or gets stuck. the lenses are still usable in case you would like to open the lens and lube it up properly or you simply can support the lens while autofocusing, with your finger. The lubricant amount doesn't fit the physical requirements (kinetic powers, gravity, friction) and it simply stops to work.
It's nothing electrical or anything complicated, this answer is valid for basically all of these:
"Hey, I finally found THE REASON and a temporary workaround: I bought this lens recently once and was facing the same problem - the autofocus wouldn't work. As my piece had some fungus and I'm quite of a noob into optical gear, I simply thought the fungus is hindering the autofocus at corresponding with the light correctly, so I took it apart using only a screwdriver and actually wrecked this whole thing but after looking at the optical tube inside which felt very dry I came to the conclusion that it would be stuck because there wouldn't be enough lubricant and the power delivered by the camera simply wouldn't be enough to make it move correctly, the angle of the optical part towards the sensor/film wouldn't be exactly 90° anymore.
So, yesterday I have bought the Exakta counterpart which is basically the same lens, I bought it without a lens hood but I still would have the hood from the Sigma - without hood, the autofocus works perfectly. Today I tried it out with the hood, and it showed the same symptomes and it just clicked - the hood applies weight on the optical tube so it kind of gets stuck because it's becoming too heavy (and the angle isn't correct anymore), this can be circumvented by supporting the optical tube, which drives out of the lens when zooming, slightly with your finger. I would be very happy if anyone would be able to find a permanent solution for this, I'm fine with taking the lens hood with me and only putting it on when necessary and then focusing with the slight support of my finger but out of convenience I would prefer to make it work normally because this is a very banal problem as it seems. Probably taking it apart and lubing the optical tube again is the solution but i'm not going to take this thing apart again, lol. Are there lens hoods for this thing which would be slightly lighter or something like that? This product wasn't going through a long term usage test in complete configuration, obviously."
You can give this hint a repair-man (lol) and he should make it work again, also you can try to install some kind of a supporting bar which would relieve the optical tube.
The thing with these lenses that they completely lack ball bearings and that makes these ridiculously prone to such faulties, it's basically the tube starting to "hang loose" because there would be too much function relying on a completely outdated mechanical principle while using sub par materials.