I have no idea about the PlusNet ticketing system, and why the ticket would have closed.
Unfortunately, your father-in-law's case has reached the 'difficult to fix' stage. If it is REIN, one or more sources of radio frequency noise are getting onto the wiring - and the source(s) of that noise are almost certainly outside Openreach's direct control.
Depressingly, there is a fair amount of equipment that fails to comply fully with the EMC regulations, but most users are unaware of this and some don't care even if the effect of their defective equipment is pointed out to them. One cause at this time of the year is flashing Christmas lights - but 26 October seems a bit soon for those. It might be a faulty power supply in some equipment.
I expect the Openreach engineers have suggested this already - but what happens if he turns off all other electrical equipment other than the router, then turns on just the router? I really do mean all other equipment, other than anything with safety of life consequences - it's only for 5 minutes. If his router synchronises, he has a process of elimination to carry out at his house, which is the best possible case as fixing the root cause is in his power. If his router fails to synchronise, the source of the noise is likely to be external.
The Kitz advice on REIN
might be worth reading, though it's not always easy to track down REIN with an AM radio.
There's only so hard that an ISP and their customer can push on REIN issues. Especially if your father-in-law is the only person affected, Openreach and BT Wholesale could go down the "too hard to fix" route and offer to end the broadband contract without penalty.
So long as PlusNet keep as much pressure on Openreach as is reasonable, moving to another ISP won't help. The work done so far indicates the problem is not just with your father-in-law's current pair of wires, but equally affects available spare pairs. Whichever ISP he chooses will use the same troubled pair, unless he has access to a non-DSL service at reasonable cost (cable, satellite or mobile).
I'm sorry it's not better news.