I had a similar problem with a different bike not a GS obviously, back in the mid 80's. I was blasting around Daytona during bike week and my bike was just running horrible as if it were very lean, (because it was) yet I could not find the air leak even after spraying all the known joints, such as carb to manifold gasket, and manifold to head joints, with WD40, no dice. I even changed the seals from the manifold to the heads which is not a fun job in a campground, and still no luck.
I happened on the worn throttle shaft mostly by accident as I stared down the throat of the barely idling carburetor and revving the throttle watching as the butterfly moved and the gas sprayed in. Suddenly I noticed the throttle shaft and the butterfly literally dancing around, as I let off the throttle and the engine returned to idle. This was a highly modded motor and the pulses in the intake track were violent to say the least. I had no parts available to fix this carburetor so my fix was to put a glob of the stickiest molly grease I could find, over the end of the throttle shaft. I carried the little tube of grease with me all over Daytona for a week and refreshed the gob every 50 miles or so.
The fix back at home was involved because not only was the brass throttle shaft worn, but the hole in the carb body was actually egg shaped as well. I bought a new throttle shaft and an oil-lite bushing to fit the shaft from McMaster Carr. Then I had to drill out the carb body and press in the bushing. A little bit of trimming and fitting the bushing and shaft, and finally the air leak was gone. You can get a new throttle shaft for your carbs and hopefully that might fix it, but check the hole in the carb body. because you'll only want a couple thousands of an inch of clearance between the shaft size and the hole size or you are right back in trouble with the air leak. The problem with the vacume cap fix is that I don't think you can get one on the spring side of the carb. Good Luck