I found out for firefox the message only appears when I'm using private browsing mode or set it to never remember history. Not sure what's up with chromium though. These errors apply when running The Question on this page as well.
I'm on windows 10
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Firefox in private mode indeed refuses to save games (which makes sense, because they would be lost when closing the window), I plan to detect and make that message clearer.
The anti-virus block is annoying (but I'd be curious to know if Microsoft also blocks its own chromium-based Edge browser.) Is there a way to report this as a false positive?