I would first be suspicious of air leaks; most bikes in the 2005 year range will have the original carb holders; the rubber things that connect the cylinder head to the carbs. These get hard with age and have a tendency to seal poorly. The clamps may also be loose. If the exhaust has been changed, I am sure there must be some crush-type gasket and it may not have been changed. Air will get directly into the combustion chamber if either the carb holders or crush gaskets leak. You can easily check the carb holders by spraying the area with WD40 while the bike is idling; you will get WHITE smoke coming out the pipes and the idle will change. The WD40 actually evaporates fairly quickly if you don't puddle it. The last thing to check is the smog system... they invariably use reed valves which are opened by a vacuum switch of some sort. The problem with this system is the vacuum hose from the carbs to the valve; if it is compromised (leaking), the bike will backfire as described. The last thing I can think of are the vacuum caps. These cover the vacuum fittings on the carbs and if they are rubber they deteriorate and can leak. Replace them with vinyl. As to the carbs, after checking for leaks, yes they can backfire if the pilot circuit, especially the slow jets are gummed up. The hole where gas flows in the slow jet is pretty small and if the bike has sat I would suspect changing the slow jets would be in order. Don't bother try to clean them as this requires using fine serrated wires which tend to enlarge and oval the beveled hole in the jet. Spend the money for new jets. You can check the flow of these jets by removing and spraying carb cleaner through them. The flow out should be fairly diffuse and in a cone-type pattern.
Last, by some reports from my customers, Stabil doesn't work as well as it once did due to the ethanol in the gas. You need to change your gas filter more often as the ethanol also eats the rubber seal in the gas cap. There are alternative products to deal with the ethanol but I haven't tried any as I ride year round.