Are the spark plugs carbon fouled? If you don't have new or know good ones to try, swap them for the ones in cylinders that do get hot.
If you are getting gas in the exhaust (I suspect it is really only condensation), gas can only get there via the piston chamber. If that is filling with gas, then the oil supply is also diluted, making very high risk of engine mechanical damage. So, check the oil. If too full or smells like gas, new oil is a mandate.
Anyway the gas can only get to those points via the carbs and the most likely culprit is the float valve system. This is either gunked up, worn, or having rust particles block the valves closure.
Does your rider know to turn off the fuel tap when parked?