Car: Honda Accord Cl7 2005 (EU)

I have this issue where the battery light doesn't stay lit when switching the ignition to ACC. Another symptom is that the fuel pump only primes for one second, and not for two seconds as it should. When the battery light doesn't stay lit, the car is not allowed to start, although it cranks fine. Switching the ignition on and off a couple of times suddenly allows the battery light to stay lit and it then allowes the car to start without any problems. I have eliminated the fuel pump relay(PGM-fi main relay 2), the PGM-fi main relay 1, the ignition switch, the immobilizer, key battery and battery itself. The problem seems to be somewhat temperature related, making it worse when cold weather. If i jump the contacts on the PGM-fi main relay 1 when the battery light is lit, im allowed to turn on / off ignition and start / stop the car as usual. This indicates that it's related to something powered by the PGM-fi relay. Does anyone know if the PGM-fi relay powers the pcm/ecu? I have considered leaving it jumped, but I guess it might drain the battery over night.

What it looks like when it is not allowed to start:
https://www.icloud.com/photos/#0ioN8...jOcaoNKy2aC75g

When it is allowed to start:
https://www.icloud.com/photos/#0N8oI...kq80-U5u-xVH2w