Websites forget the user was already signed in when launching a browser through Steam Deck game mode

[UsernamesAreNotMyThing]

Your system information

• Steam client version (build number or date): Most recent stable update (5/24)
• Distribution (e.g. Ubuntu): Steam Deck LCD
• Opted into Steam client beta?: No
• Have you checked for system updates?: Yes
• ~~Steam Logs: [generate by running this command in a terminal tar -zcvf ~/Desktop/steam-logs.tar.gz ~/.steam/steam/logs]~~ May not be much help here
• GPU: AMD Custom GPU 0405

Please describe your issue in as much detail as possible:

Websites don't remember the user was previously logged in when launching an external browser in Gaming Mode, and behave as if you were never logged in in the first place. This is especially apparent with Google login, where it thinks I'm using a new device every time. I can close and relaunch the browser as many times as I want without ever encountering this issue in desktop mode, including through the Steam client and with Gamescope from the command line, but it occurs immediately when launching it through Steam in gaming mode. And it doesn't seem to matter if the application was installed through Flatpak or not.

Steps for reproducing this issue:

1. Log in to your favorite website in a browser in desktop mode.
2. If you haven't already, add the browser to Steam as a Non-Steam Game, and verify that it runs.
3. Log out of desktop mode
4. Run the browser in game mode as a non-Steam product.
5. The website behaves as if you have never logged in on that device before, which is false.