Although this is not burnishing, perhaps a hint from restoring brush finished or non-reflected finishes on stainless steel will help. Many watches are finished this way. I’m often asked to take out small scratches.
I sand using a small piece of 1000 or 1200 grit 3M polishing sponges held in a locking tweezers and lubricate with sewing machine oil. Depending on the piece, I sometimes use a dish washing detergent and water mix to lubricate. I think both of these lubricants would work equally well when burnishing.
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