It's pretty simple really. If you have fuel, compression, and air it will run. You aren't losing air or compression obviously so you have to be losing fuel. The fuel shutoff solenoid bolts to the pump correct? How are you determining that it can't be dropping out? Power to it doesn't mean it can be a weak coil. Looks like an electric lift pump is used. How do you know it's maintaining sufficient pressure after start up? It is one thing to build up pressure before starting but it might be failing after start up. saw it on a John Deere engine more than once. Bosch fuel system. Variable flow lift pump. It would be running full throttle and die. Pump was failing and when the pump stalled the engine died. On this system fuel pressure was maintained by the ECM lowering or raising the voltage to the pump to maintain the desired pressure. The pump would start again if I tapped on it when the engine would try to die. I