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If it does have the metal type plugs caps, as the others say, they never really worked when new....nearly everyone with any intention of have the bike run reliably used NGK, two straight ones in the middle and two cranked ones on the outside cylinders.
Sounds very similar to a problem I had with my zxr 400, in fact the symptoms are identical, it was float heights, when the engine was cold it would run fine, after riding for 5 minutes or so it would have the symptoms you describe, the only thing that was changing in that time was engine temperature, this lead me to think that it was over fuelling (running rich) , so by trial and error I adjusted the float heights, I couldn't just set the floats to the factory settings because the intake, carbs and exhaust were modified so the standard settings weren't necessarily the correct settings, at the 5th attempt the bike ran perfectly, so I guess the question is after you balanced the carbs and it ran well for 10 mins did you subsequently let the bike cool to stone cold ?, and did it run ok for 10 mins after that subsequent cold start up?, I'm absolutely sure it won't be carb balance btw, cheers Mark.Another thought would be a vacuum forming in the fuel tank due to a blocked breather, not so likely but easy to check, just go for a ride with the fuel cap off (well life is no fun without a little danger is it!), this happened on my XJR1300 a few weeks ago, after a few miles of imitating John McGuiness the engine started to cut out, I opened the fuel cap and there was a huge rush of air into the tank, I got home by keeping the revs below 3k and found that the breather hole in the cap was blocked, no problems since cleaning it out, I doubt your bike has this problem but it's a quick check.