Attaching zipper to character

I’m trying to attach this zipper to my characters cloths. As you can see, it’s not behaving as I would expect or hope for.
I tried pretty much all methods I could think of. Parent it to the cloths, parent it to the bone, vertex etc.
Actually I’d just want it to stay in one place, without deforming. The deforming is what bothers me the most. So any idea?

An option is to attach the zipper to a separate bone parented to a deform bone. That should give independence.