Have you tried removing the drain plug on the bottom of carb #2? That's one sure way to determine if that carb is getting fuel at all. (Have a container handy, of course).
If there's fuel, then that particular carb has to be plugged up and needs cleaning. If there isn't fuel, then its inlet screen could be plugged up, or for some reason its needle valve is stuck shut. I've never personally seen this happen... but the cause of no fuel should be easy to determine, only because it's so simple. Clearly, fuel is reaching carb #2's fuel tube because cylinder #1 is firing OK. (Fuel has to travel through #2 to reach #1).
We are assuming that you've ruled out ignition? You've swapped wires #2 and #3 with no difference noted?
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