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fine sandpaper might work, but NO steel wool that may lead to rusting, if you do use sand paper, when done give a good scrub with the BKF again it will help the passificaion of the newly exposed SS.
Since it looks like it may be organic stains, if you have a homebrew shop in the area a good soak in hot PBW (Powdered Brewery Wash) might help. Last ditch effort would be some professional strength caustic used in breweries to clean all the SS, but is very very (oh hell one more) very dangerous if you don't have experience using it.
The caustic, and the PBW are made to work on organic "stuff" and in time will break it down.