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Conking out after 15 mins then not starting till cooled down sounds like an electrical component failing at higher temperature... coil maybe. Or it could be fuel starvation but unlikely....
Could be vacuum in the tank caused by a blocked breather. Take the fuel cap off when it conks out and if air rushes in it could be the culprit.
Not too rich is it? Just chokes it when warm possibly.What colour are the plugs? Do they indicate rich setting or the opposite? If rich, crank it with throttle wide open to get more air in and it may go. There are breathers into the float chambers of each carb that allow the fuel level to stabilise, you can run into problems with filling if these are not all clear.As a means of diagnosis, when it stops you can open the throttle and spray some easy start into the carbs via the filters to see if it'll fire again straight away.
This kind of fault can be difficult to diagnose.... if its fuel starvation and you go through all the various components to ensure they aren’t blocked or damaged you will eventually eliminate it. For an electrical fault it will also be a process of eliminating the faulty component. If an electrical component is damaged, cracked, broken etc as it warms up it can lose connection then as it cools it reconnects.... so the only real way to diagnose it is to replace components one at a time until you solve the problem. The condensers are so cheap they should be changed anyway, if that works then its a cheap result... i would suspect the coils but it could be anything. Does it stop dead or begin to missfire and die gradually?
The gradual thing doesn’t sound electrical to me..... I have new ds condensers on my k4, one of them failed after 50 miles (missfiring and spaarking like hell under the points cover) so I can’t recommend those... I’m wondering if its fuel or worse still mechanicsl. Have you checked valve clearances and compression?