I cleaned out everything in the float chamber of my carbs and it was still hard to start. The choke, or as Honda calls it the bystarter valve, had to be cleaned. The problem is that each valve faces another carb, so they had to be completely dismanteled from each other. I think there were 36 phillips-head screws holding the carbs together, and each wanted to strip their head.
After lots of cusing, impact wrenching, and vice grips on two screws, I got them apart. I cleaned out 14 years of crud in the bystarter valve, without mixing up any parts. $250 would have been cheap for what took me two whole days. Starts right up now, using the choke when cold. I never use the choke when the engine is warm.