My knowledge of GS's would fit on the back of a postage stamp, so bear with me on this suggestion.
My first BX a 1.6 litre, used to do V8 consumption figures when I first bought it. It also was owned by a well known Cit specialist so you would have to expect it to have been set up correctly and that was a problem, trying to find out what he'd done and which I found was mainly things most of them seem to do which was dispose of many of the parts associated with the fuel/air intake side of the engine. Usually done I feel in stages firstly possibly due to slack workmanship which creates a problem, then from there downhill all the way as they 'modify' things rather than restore them trying to sort it.
Amongst the things I restored on it was the idle jet which when checked was found to be a 70 when a 45 was the original size. The main and compensater jets were fitted assabout, the float level was too high and the most important piece of handywork to be discarded was the hose from the hotbox on the exhaust that fitted to a 3 way thermo flap controlled diverter that was designed to allow the heat from the exhaust manifold to be drawn into the intake but which thermostatically was diverted once the air in the box hit a certain temp. The manifold to diverter hose was missing, the flap in it was held wide open allowing ambient temp air, regardless of ambient temperature, to flow into the throat of the carby.
A condensed version of what I did was this: Fitted a 45 idle jet, reversed the main and compensator, set the float height so it would idle and rev out when hot (cold engines are only that way for 1 % of running so who cares if it's rough when cold) by trial and error rather than pre determined measurements, reconnected the hotbox to the diverter, removed the screw jamming the flap so it again became thermostatically controlled.
The reason the latter was so important as I discovered, was that when I accessed a service manual on this Weber carb (and I have no doubt this would apply to others) it seems the initial settings on them are done in a rather strange way. There is a setting done that is proportionate to ambient temperature. Technically speaking, this will not change dramatically if an artificial ambient temperature is established via the hotbox to thermo flapped diverter is used, however, when the "Australian conditions" mentality clicks in that tells these guys that 30 degrees here is different to 30 degrees they get overseas, this gets all discarded. At this point, the setting within the carby (and no, I'm not now talking about automatic chokes before anyone asks) :nownow:

compensates for the temperature difference by bringing into play what they refer to as an "enrichment device" possibly controlled by a bi-metal strip, that richens the mixture to allow for what it thinks is cold driving conditions. If the carby is retuned and this 'throttle' as it is referred to is reset to allow for the lower ambient temperature, then this may not be quite as big a problem, but whilst ever the hotbox and diverter are disconnected, this enrichment device will play a constant part in affecting fuel consumption. When mentioned on an overseas board, it seems that even over there, the average greasemonkey discarded the hotbox hose sometimes even under warranty with resultant fuel consumption problems.
When my car was sold, the new owner was rapt in the performance and consumption but a few weeks later rang to have a whinge saying it had been taken to a different but equally well known specialist who gave them a right old sermon about the tuning of the car including showing them how he'd "modified" the air intake, reset the float and put the jets back the correct way around, but didn't know what I'd done because he couldn't get it to run or idle as smooth and the fuel consumption had almost doubled to what they were getting when they bought the car at first.
I didn't waste any breath trying to explain. Just referred them back to the expert. :crazy: :crazy:
The point I am making, is that if the original set up was a similar set up to this BX, then this may be one avenue that hasn't been explored and might be the answer you are looking for but slipped through the net if the removal of these pipes from the hotbox is as common on GS's as it was on the carby BX.
Being an air cooled engine, these kinds of things I would imagine could play a critical part of correct tuning and overall operation.
Alan S