By chance is it a NEOPRENE cinch? I was leasing a horse who had a neoprene ENGLISH style and I had a brand new Weaver Smart Cinch NEOPRENE one. It rubbed all the hair in one patch in just a few uses.
Ran out and picked up a 100% MOHAIR cinch (Wrangler Twenty X with a buckle roller -- so it's smart cinch style, too!!) and it totally took care of the problem.
Hope it's not a medical issue, but hope this helps!