I think that will be the the pilot jet as it controls about 25% of throttle and obviously when your chokeing it your adding fuel thats the jet on its own not the main jet on the emulsifier tube .but I would have a good clean on both as a reference point and obviously the easiest route is a good clean through.
I was out the other day and my bike started playing up, giving a bit of throttle and would die, then it would go again and then die again.
I pulled the pipe of the emulsifier and a load of white oily water came out, I left the pipe of and revved the bike a bit and more stuff came out. When it had stopped I put the pipe back on, and the bike ran fine again.
Doesn't sound the same as your problem, but thought I would share.