ThanksWhen you press the starter button, a feed also goes to the fuel pump power relay, through a diode, and then back out to the fuel pump. Once the start button is released, its a pulse from the ECU to the number one coil that also goes to the fuel pump power relay, to tell it to provide some power to the fuel pump for a short time. This is so the fuel pump is not constantly working.
So chances are you either have a fuel pump problem, or a fuel pump power relay problem. Change which ever is convenient, or open the fuel pump up, and see if you can see a problem in there, sometimes the contacts in the pump can get worn or welded together. I'd check the pump as the power relay is a sealed unit.
I have a spare set of coils and ecuI would imagine your losing the feed from the coil to the fuel pump power relay. This could be a bad ECU or coil. But check the cheaper wire option first.