Have you tried deleting the Steam plugins folder? I've never had this issue so I can't guarantee this'll work. Do it at your own risk, I am not an expert but I have had my fair share of issues with Galaxy so this is just a possibility.
It should be located at "C:/Users/[your Windows account name]/AppData/Local/GOG.com/Galaxy/plugins/installed"
You should see a folder for the Steam plugin; it'll say "steam_" followed by a random string of letters and numbers. You might also see a configuration .ini file.
I would not delete either of them, but instead, cut them from there and paste them to a temporary location, just in case. After you do that, start Galaxy again and check. If it worked, if the integration was disconnected, try reconnecting and starting the extension again. Corruption of the integration is probably the cause so this might help. Again, I can't guarantee anything though.
If that doesn't work, you could try a full reinstallation of Galaxy but that can be a pain as there's currently no way to back up your side panel bookmarks or settings. If my idea doesn't work, you'd probably be best to contact GOG support. Every time I've had an issue with Galaxy that I couldn't fix, they helped me figure things out and were extremely friendly and had a pleasant, personal vibe. Just a tip, though: if you contact support and it takes a long time for a response, don't give up. Just add a new reply to the ticket or PM a blue on the forums and that should help. They have a very small support team so they tend to get overwhelmed, but they're all awesome.
