You will wind up with a polished area where the turn ring is/was - and the grain won't match the rest of the cylinder. Thus, my comments about using something slightly more abrasive than polish to (to some extent) remove the actual turn ring while keeping the "grain" in the finish ...
There needs to be a flat aspect, for the lack of a better term, to you work on the ring - otherwise polishing it with, say, your finger you will simply deepen the ring itself. A flat nail file, although fairly soft, will enable you remove metal around the ring - but not in it. It's OK to use a Revlon nail file - no one will accuse you of trying to be Kaitlyn!! ...
I've done this dozens of times - it works - otherwise I wouldn't post it!!