Every new conversation in your Engage Inbox is set to ‘Unassigned’ and ‘Open' by default.
To close a conversation, simply click on the check mark on the top right corner of your message window.
That will set the status of the conversation to 'Close'
You can also use the Keyboard Shortcuts in Engage Inbox to close the conversation.
The shortcut to ‘close’ a conversation is ‘y’.