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Chain only runs cleanly when I keep the shifter pressed
(sh.itjust.works)
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If you can make it go away with the shifter, I would guess it is not the limit screw but that cable tension (I don't think the shifter would help alleviate the problem if the limit screw is set too far in). You could start with using 1/4 turn increments of the barrel adjuster at the shifter. Go slow. Do one 1/4, shift around and stop back on large cog and see how it sounds and looks.