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2 things to consider.
If the linen is not broken in, being a stiff coarse style & the razor edge has been taken to an overly fragile condition you will degrade the edge.
Leather will not but you...

Never used oil on my Kanayamas. Suspect a radical draw change if you do. The damp rag works a treat .

For softening the linen I used a steel brush on mine. It doesn't destroy the linen as you...