I'm a librarian and I have to put in a few hours on the virtual reference desk. We use RefChat. When I need to move to another tab group i get a message asking if I want to leave or stay on the page. If I choose leave it logs me out of RefChat. If I choose Stay it opens another instance of RefChat on the new tab group. It does the same thing with our Help Desk software. I'm using Firefox 57.0.

Love this add-on...I can't live without it...