Considering that A). it did use to work, and B). you seem to be the only one with this issue, perhaps you should figure out what changes you have made recently to your site and especially what add-ons you have installed since the time it used to work.
Conversation Manager does not require any edits to the 'conversation_list_popup' template so if you edited that one then it must be for a different add-on.
Do the TMS edits show up as Green in colour?
If not, then that means your templates were previously edited and so the TMS cannot find the correct lines to replace.
If that is the case then you will have to do the edits manually.
I like this a lot, but am working on a minimalist forum. Is it possible to use the Conversation Participants parts, but disable the Conversation Management part? The pop-up in conversation management doesn't suit the minimalist feel. I'm able to switch from avatar to text on the participants, which is great!