HTTP these days (via WebSockets) allows for the creation of stateful connections. My idea is that HTTP will just expand to encompass any remaining non-HTTP use cases. HTTP isn't even coupled to TCO anymore!
The WebSocket Protocol is an independent TCP-based protocol. Its only relationship to HTTP is that its handshake is interpreted by HTTP servers as an Upgrade request.
If you do connection setup, service disambiguation, redirects, and so on with HTTP and switch to a non-HTTP interactive protocol, what you've effectively done is create an HTTP sub-protocol, and I don't care what the document you quoted says.