Only suggestions to your end...perhaps worth to try
If you have the modem near by a tv or speaker or any electric device etc, remove them far, also if you have it at an extension electric socket, plug it direct alone
https://support.zen.co.uk/kb/knowledgebase/broadband...
That may helps to have less noise stress at your situation
Change rj11 cable from phone socket to modem ... even if is sealed or grounded i think may still cause issue by trapping additional noise if the rj11 port at modem is not grounded, so try with an ordinary rj11 cable
i hear a BT 80B RF3 Junction Box works in some cases even if is for different purpose (i think needs to be installed before the master socket & i know you have tried filtered plate but it may makes difference because contains coil so more realistic in your case?) and also try this time 2 filters in line (connect them at the phone indicator)
It will not fix the issue if is from isp /line but it may makes little better the situation
Edited by deleted (Mon 20-Jun-16 09:42:21)