No. Desktop notes use local storage, but hosted AI, connected services, and cloud features need network access and can send relevant information outside the device. See Timeline and privacy.
Check the current chat list and Settings → Archived chats. Your personalization settings can archive finished chats. Closing a visible chat and deleting a note are different actions.
Char is in private beta, and the handbook focuses on desktop. Check your app version and use Settings search. If the control is still missing, contact us with the version and platform.