They aren't seeing any issues now but will have put that down to the elevated target margin.
I've just posted a link to a line stats graph in response to ian72's post. I don't know whether you'd consider the SNR stable or not.
Now that we can see for sure that there is no ring wire connected there is not much that can be done to improve things. "Prowling around with an AM radio" may or may not show some local noise sources but it isn't is going to produce a solution because nobody is likely to re-route the cabling or be able to fix the noise source. In any event as the noise margin is now apparently static I doubt anything will be found.
IMO the reason for the elevated margin is made pretty clear in the first post and in the clarification I requested a bit later...
I've recently had to replace my long-serving Linksys router as its progressive failure was manifesting itself as increasingly intrusive phone line noise. I'm now getting good service from a modem and separate router (both D-Link, new), but the speeds are somewhat lower than results I was getting a year or two ago.
A dodgy router lead to instability. The margin was elevated to compensate. The router was replaced and the OP confirms that the audible line noise has gone. The obvious fix is for the ISP to reset the margin to its default 6dB. All that is needed is to find a support tech worthy of the name who can understand that simple senario.
Indeed, rerouting or rewiring is very unlikely. However, the ISP person that I spoke to was knowledgeable, informed and helpful; even so, she was quite clear that there was evidence of 'interference', although perhaps I should have pressed her a bit harder about its nature. That is with the new router in place, which is in the same location as the old one.
In the light of the recent feedback from this forum, which has 'eliminated some possibilities, I'm coming to the conclusion that the location of the router close to the mains distribution board and the central heating controller could be a significant factor. To that end I'm investigating relocating it, but that will not be straightforward.
This is all good stuff - while it's not straightforward, I'm feeling cautiously optimistic that an improvement is possible. Thank you!