Dear Kleinstein,
Thank you for your answer.
About the error 608, I made a mistake, it is 609 and 610 that are showing up.
The text is corrected.
The error listing from the previous attempt of fixing it has resurfaced from the notebook, there was the same as in this post:
https://www.eevblog.com/forum/repair/hp-34970a-self-test-fail-ideas-needed/msg942341/#msg942341Errors 609, 610, 611, 617, 618, 621. Changing U101 fixed 617, 618 and 621.
Today I tried to replicate the Selftest measurement that WanQi did in the above discussion and that you mention (I found Pin6 of U106). I was able to see the initial voltage changes but it was not sustained, there was just a peak. This is so strange, I must have made a mistake on the setting of the scope.

It looks like your advice is what should be done: "check the amplifier output (ADIN signal, e.g. at pin6 of U106)
with a shorted input and in amps mode."
Can you explain to me how would you do this?
Then, I started to verify the voltage rails on the DMM board, but I was not able to see any....

Finally time ran out and I had other priorities to fulfill.
Unfortunately, I have to go for 2 weeks to another location for my work. So I will resume the checkups when I come back.