Currently I am using a fork of the official widget called “Web Browser Extended” for this functionality, but it is kind of hit or miss. Sometimes it will keep me logged into whatever website I have it set to and sometimes it won’t. Also, only audible notifications come through most of the time, Google Messages appears to be the only site I use that will actually push a notification pop up to the desktop. I think the devs at KDE could probably implement this much better and it would be a very useful productivity feature. I use the widget for Google Messages (texts, synced with my phone), Google Chat (work), and Gmail (misc).