All the four cylinder goldwings are EXTREMELY sensitive to carb sync, so I think you DO need to sync 'em. But, that said, I doubt it's carb sync that's causing one cylinder to run rich.
Valve adjustments are simple to do on a goldwing, so that shouldn't be too much wasted effort.
I'd be inclined to do a warm compression check first, looking for rings, etc.
Still, it seems unlikely that ONE cylinder would run rich from worn rings. I mean, if one is worn out, chances are that all are tired. I'd still look for a carb issue. Or possibly a weak ignition / plug / coil / plug wire on that one cylinder.
Kirk