Nothing causes more grief in a 5440B than those relays, even when they look pristine. I would plan to replace them all.
I have seen quite a few painful stories about the relays ... if my PCB/off-the-shelf relay approach works then I'll plan on doing a complete replacement ala @wn1fju.
In the meantime to allow me to get a bit further with fault diagnosis I've figured out a reasonable routine to get the existing relays cleaned up a bit, I'm breaking all the rules, but it's a choice between just throwing them away or having a go!
Bear in mind that almost all of the relays I've looked at so far are not even able to move enough to make physical contact when applying 5V, let alone low resistance electrical contact.
So I give them a 2 minute clean in my ultrasonic cleaner but in a small container of IPA ... this is generally good enough to start them properly moving again. At this point there are usually some good contacts and some bad ones.
I gently scrape (caress?) the relevant contacts with a scalpel (although I seem to spend more time cutting myself at the moment) and repeat until I get sub 1R resistance on everything.
Then I setup my PSU to do 3 second on/off cycles and monitor the resistance as I've noticed a few creep back up again if they aren't properly clean.
If all else fails then a squirt of contact cleaner and another 2 mins in the ultrasonic/IPA seems to do the job. It's a bit painstaking but it seems to be getting there.
Probably not a good thing to do to ensure long life, but as per the above these things are a nightmare anyway!
Photo attached ... three done so far!