Some things with GS' that can cause issues here...
Are your valve clearances in spec?
Have you balanced the carbs? (Do that after valve clearances not before)
Looks like pods and potentially a free flowing exhaust, which is likely going to need richening up possibly significantly.
On my Katana, my exhaust isn't all that free flowing but I have the stock rubber velocity stacks with Ramair sock filters and I still had to go up a few pilot jet sizes before it would idle nicely, symptoms exactly as you are saying (when I had stock carbs on it anyway).
Once the mixture screws are wound out beyond about three turns you've maxed the adjustment out and need to increase pilot jet size.
If you keep the choke open and crack the throttle, does it still die?
Have you been able to run it long enough to get properly up to temperature, and if so does that make it better? They run richer when warmed up.
Have you adjusted the needle at all? If the needle is still in the stock location and slides lift, it could be the needle jet that's too lean, and adjusting pilot and main jets won't help with that.
Also, how is your plug colour? I'm not great at reading plug colours myself, but if you start with fresh clean plugs and do a plug chop you'll see if it's really lean (white) at least.
Caveat on the above is I'm no guru, that's just my limited experience from my 82 GS450 and 84 Katana.