Thanks but that's not what I mean.
If I don't leave the stream tab in my browser open the chat content vanishes and when I open a new stream tab the chat box is empty.
In other words, if I shut down Chrome and then re-open Chrome and return to my stream the chat is empty..
Hate to say it, but I dont think you'll be able to fix this.
Twitch chat is browser based, and if you were watching a twitch stream from your browser and had to reload the page - the chat clears -
So, obs is doing the same thing, because its being added as a browser source - especially if you have it set to [shutdown when not visible & refresh browser when becomes active]
You could attempt to turn those 2 settings off, and see if it will hold the session open for you - but you are adding a lot of workload onto the computer if you do that because they will still be running in the background if you aren't on that scene.
Another way around this is to try to not add the chat as a new browser source on each scene. Instead you could try to create a nested scene with your background & chat browser source - and then on each scene you could add the nested source which would carry the same browser source from scene to scene.
Hopefully this helps give you some ideas, if not - let me know.