I don't think you can store the cache on client given the thinking is server side and you only get summaries in your client (even those are disabled by default).
If they really need to guard the thinking output, they could encrypt it and store it client side. Later it'd be sent back and decrypted on their server.
But they used to return thinking output directly in the API, and that was _the_ reason I liked Claude over OpenAI's reasoning models.
