Mike, the slide could stick. I regularly take them out and clean 'em good with wd-40 and also clean the slide bore too. Anyway, you can take the slides completely out and it will idle just fine. You could just stick a long screwdriver down through the airbox to push the slides back and forth. That'll give you a good idea if they're binding at all.
As for the coasting enrichener, think of it as a smaller version of the piston slide/diaphragm. Instead of making the main carb bore variable, it opens/closes off the pilot air jet #2. At idle or when you close the throttle during engine braking, the piston closes off this jet thus making the pilot mixture richer. Starting at off idle, a small vent for this piston creates negative pressure and opens up PAJ2 thus leaning out the idle mixture. The idle mixture comes into play up to 5000 rpms. Air from the pilot jet #2 combines with air from pilot air jet #1 to mix with fuel from the pilot fuel jet. When you open or close PAJ2, you're effectively leaning and richening the pilot mixture.
Next time you look at the top of your carbs, notice the little vent. It's the oval hole. When the throttle is opened, air is sucked into the vent. Some air is used to open the main slide. A smaller portion is used to open the coasting enrichener plunger and another small portion flows through PAJ2.
If the enrichener is stuck closed, then you'll probably run rich in that cylinder. If it's stuck open, the bike will pop at deacceleration or even at idle.