2009 F8GS - not exactly the same symptoms as yours, but idle speed related.
Bike started to "hunt" a bit, with the revs coming down from the normal idle level, then moving back up, all without any throttle input. Start to twist the throttle and all seems right, so it was just at idle that the revs wandered a bit.
Opened up the airbox and found oily water sloshing around the bottom of the airbox, at a level that brought it upto the level of the idle stepper motor that sits on the right hand side of the airbox, not far from the bottom.
The water was condensed vapour from the engine breather. I dried it all up and gave the idle stepper motor opening a squirt of GT85, hoping that it would sort out the idle. NOPE. I ended up having to pay the stealer to re-set the idle stepper motor - a 5 minute job, but still silly money for 5 minutes. Bike was fine after that.
The airbox on these machines does not have a drain valve of any sort to allow the draining of said condensed engine breather vapour, unlike almost every other bike that I've owned!
I ended up corking up the inlet into the airbox that the engine breather connected to from the outside and directing the engine breather, via a oil resistant hose, into a small alloy drinks can located in the home made sump guard I had fitted. Worked a treat.
Now a days, on the latest F8GS and F650GS twins that I've had / got, I just make it a habit of inspecting the inside of the airbox once a month in the winter - I ride everyday and the colder months seems to generate more condensate in the airbox, drying out as required. the warmer months do not seem to be a problem in this respect.
Never had the same problem re-occur in over 8 years of riding these bikes
Just my mutterings.
SteveT