What’s causing my GSX-R600’s poor fuelling?

My brother’s got a problem with his GSX-R1000 K6, the symptoms are: No fault codes showing; misses when held between 3-4000rpm; only does it when hot; all four plugs sooted up; no problem accelerating; strong petrol smell.
He’s quite mechanical and has checked a few things like the plugs, hoses checked for air leaks, fuel pump is clean, sensors unplugged to check if they register a fault code.

With a strong petrol smell and sooted up plugs the engine must be running rich on a partial throttle. It’s unlikely to be a blocked air-filter, more likely to be the fuel injection making it run rich. The low-speed fuelling takes its cue from the inlet manifold vacuum. If it’s reduced so the closer to atmospheric pressure that will make it run rich.
There are four pressure sensors whch are little square sensors attached to the airbox with four pipes going to the inlets, s give them a good check over. It’s also possible the valve timing is out by one tooth, it’s not enough to clip a valve, but if the cams are opening at the wrong time, there’s less vacuum and the bike runs rich.