'95 5S-FE Randomly Shutting Off Options

[Chockomon]

Hey guy, I'm not too familiar with Celicas as I'm more of a VW guy, but I would love some help on some issues I'm trying to diagnose. My friend just bought a 95 GT convertible. The engine runs great and it drives amazingly. However, it has this random stalling/shutting off issue and I can't seem to figure it out. Here's what happens:

The car will shut off out of nowhere. Doesn't stutter, shake, no CEL, nothing.
It happens at any time, independent of temperature, weather, engine temperature.
Usually will turn right back on as if nothing happens.
HOWEVER, sometimes it will not turn back on. When this happens, no CEL or battery light (warning lights) will turn on when the key is turned to the on position.

So where do I start? I'm thinking that the issue could be a faulty relay but I don't know which to look at / ring out. Could it be the Main Engine Relay? EFI? How would I go about checking the ECU on this car?

Thanks guys!

[Chockomon]

So I printed out some wiring diagrams and I'm going to ring out some wires/coils/etc.

Presure2 - what am I looking for when I jump those 2 slots?

[presure2]

So I printed out some wiring diagrams and I'm going to ring out some wires/coils/etc.

Presure2 - what am I looking for when I jump those 2 slots?

doing that bypasses the fuel pump relay and the circuit opening relay.
if it runs after doing that, you know one of those 2 is bad.