Hi,

Relays don't usually fail but these maybe old enough. They can go out intermittently just before they quit all together. This is only if they get too hot but not when first starting up so I'm not willing to say the relays, are bad, yet!

Let me see if I can some more light on what is or is not happening, in order to coax out some more results back, in conversation, from your symptoms.

You said it runs as expected when you bypass the relays in the fuse panel or by applying power straight to the pump circuit near the main pump?
Does this mean, if you shut the engine off, during the bypass, it will restart immediately every time?

Can you also let it set a little bit, with the pumps running and then restart the car. Does it act like a normal car that you had before the rough idle started.
It doesn't flood out or run rough after the last or prior other procedures I'm asking about?

Running rich or losing starting rest pressure on the fuel rail can be caused by a bad FPR.
Look for leakage from one side of the diaphragm or from the return line going to the gas tank.

The two relays each control the injector system and then the pumps separately as they are separate circuits turn on by the ECU.
There could be bad ground or connector on one of the relays only! Inspect both connectors and each wire terminal in both connectors, for tightness and lack of corrosion.

I have some other ideas, but we need more things to get done, in a certain order, to create a path to the malfunction.

Phil