after re-reading your post I decided that I would replace the K40 relay and see if that fixes it...it is cheap, easy to replace and can easily cause the problem you have described. Could also be the lift pump though I would think you would see the lines full of air when it stalled and also don't understand why it would start normally when cold. That's what leads me to believe it is electrical and not fuel which is causing it.

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Marty D.

2013 C300 4Matic
1984 BMW 733i
2013 Lincoln MKz