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Your float bowl is not floating properly. It is sticking. Maybe a sticking float, maybe a bit of debris in the float valve, Maybe the gelatin gremlyn in there.
I take it you were pulling the carbs from the bike each time you checked them? Reason I ask is that "checking carbs" on the bike would not require pulling the gas lines. If (when you finally swapped them) you pulled the lines and replaced, you may have cleaned out debris in the left line and now it works (with the right carb in). Swap back and see what happens. I think you have determined electrics aren't involved if the wire swap didn't prove to correct that side. Baffling yes but could be debris or something similar that finally worked its way out. Let's hope anyhow.