Can I delete my conversations?
The current interface doesn't offer permanent deletion, message by message or conversation by conversation, of private chat history between Dialova accounts. Closing a tab only tidies up your discussion space.
Close, clear display, and block
These actions are not equivalent. Closing a private chat removes it from the visible tabs; a display-clearing command shouldn't be understood as a database deletion. Blocking a person limits private interactions but isn't a button to delete all exchanges.
For a request regarding your data
You can contact the team from the Contact page, specifying your request without sharing your password or login code. Permanent account deletion has its own process in Settings → My account. Don't assume a history has been deleted until an action or processing confirms it.