I can't see any way that a tune can relate to a belt.
Belts squeal when they get old for a variety of reasons - cracks, heat, hardening, loosening up, and so on. Just simply existing and being used in a hot environment. I suppose a tune could potentially make things hotter under the hood and make them wear faster, but it's a stretch. it's not like damaging an engine from excess boost from a tune.
Belts are generally easy but a pain in the butt to replace. I've changed them on my own cars and friends have come over for the same. It ranges from a couple minutes to a couple hours depending on how things are arranged and what if anything needs to be loosened or removed. no idea for the 1-series, haven't had to do it yet.
I guess if I'm saying anything it's simply a wear item and happens on every car eventually.
Also - belt dressings are sometimes waxy, for lack of a better word, and that can actually cause more slipping. In my experience they work for a short time and the problem comes right back. It won't really go away until it's replaced and properly tensioned again. Not in my experience anyways.
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