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Can you give some more info please?
EFI or carburetor?
Sounds like the engine is flooded from one or more weeping injectors.
How long does it need to be turned off while warm?
Does it blow black smoke for a short period when it eventually starts?
I'm in Australia, and I think the phone call feature is available within the US only.
I feel I can help you without talking on the phone.
I you need to open the throttle a lot when the engine is hot, more than likely the engine is half flooded, like a percolating carburettor, from either weeping injectors or some other electronic issue.
At least you have proved the fault by opening the throttle. Once the engines starts, it performs normally for the rest of the driving cycle?
I would fit a fuel pressure gauge while the engine is hot, and monitor the pressure after shutdown. Then sparkplugs out for inspection. Let the engine sit for 20 mins. Crank the engine to see which cylinders the fuel went into.
Injectors out, ultrasonic clean and flow test, refit all parts and test vehicle.
Have you had a chance to perform the tests I have described above?
I hope my advice has been of assistance to you.