As far as I can tell there are no extensions in the house, there's only one telephone socket.
This is the inside of the socket:
Socket
I'll post back in this thread in a couple of days with any changes.
Looks like everythings OK at the back of the NTE5 - have you also tried different filters and/or filtered faceplate?
Are you insane? Thats not even got a test socket inside it... its empty just a front plate on with no SPLIT faceplate. This is an obsolette master socket, likely one with a T in a circle or a non-split BT socket with the piper man logo. I can't quite tell from the angles there but it looks like an LJU (just an extention plate as its quite thin and the socket itself recesses into the wall).
Whats more it looks like theres a lot of matter inside the socket walling (dead insects flies or dust etc) which have caused issues before especially if they are in contact with any of the wiring or circuitry. Whats more is if this is indeed the master socket or only socket in the property it still probably has a functional ringer wire, these are no longer needed as they used to work with older phones to make them ring now all they are useful for is sending interference down your telephone line into your modem.
Basically that socket needs replacing with either a modern openreach socket or an already filtered faceplate - this will require a broadband engineer from your provider.