That's possible in general but not for this application; a chat interface to an LLM isn't very useful unless you can tell it whatever you want—including GDPR personal data—and then pick up the thread of conversation later.
It is kinda possible to do store that in browser, but as I've been finding with my own browser-based front end for the API, the browsers seem to clear this data a bit more than one might expect.
Or at least, Safari on Mac clears it.