Ok. Great response. Thank you. My first impression... The bike sat and fuel dried up from sitting (without the carbs being drained for storage) and left a residue. When the bike started being ridden, the fuel started soaking and breaking up the residue and it has clogged some pilot passages and/or jets. Not saying for sure that is the problem, but I have seen this exact scenario too many times. The fact that it has developed means it's not mis-adjusted cables or linkage. And you have likely tried to adjust the idle only to find it will just now drop down to nothing out of nowhere and die. It's either nothing or 2K? LIKELY, you are gonna have to warm up to the fact the carbs need to come out. The good news... the RC36-II is one of the best carb V4s to work on. Start a new thread and get ready. Don't do ANYTHING before reading, researching, asking. There are likely no shops that will work on it (or that you actually want to let them work on it). Welcome to the wonderful world of carb VFRs!! You're not the first, so there is plenty of resources and advice to be absorbed. Thicken up your skin a little bit and don't be afraid to ask. Come in with the right attitude (like you already have) and you will get it. There are also experts like Squirrelman that do amazing work, fast turn around and stand behind their work if you don't want to become a V4 carb expert (which you will be if you decide to dive in). Seriously consider that option. If you store the bike properly when not used, you will likely never have to worry about them again. Oh... you have a Factory Service Manual... RIGHT?
without removing carbs you can check boot/clamp possible leakage by spraying WD or carb cleaner [even water] onto each boot while engine is running. any deviation in idle speed would suggest an air leak. you might try turning tighter each boot clamp with a suitably long screwdriver, maybe only a fraction of a turn.