OpenAI and Anthropic will probably now resort to save this server side, instead of relying on encription to be able to keep state on the client.
That's not a trivial thing to do for them because they offer zero data retention environments to enterprise clients.
Encryption is irrelevant here. Even if it was kept fully server side, the actual issue is that they allow starting a conversation in a strong model and continuing it in a weaker one. Disallowing that entirely would be a huge hit to user experience.