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I don't believe so. Some people feel that it improves the edge, I have never noticed a difference. YMMV.

IMO (which is backed up by an antique barber manual I read) the cloth component is on...

My linen strop is just a folded loop of linen/cotton blend.

Personally I believe that cotton or denim is much better for the edge than the coarse cotton webbing that is being used as the fabric...