Browsers should connect to port 80 and perform a GET for /tls-cert with an Accept: header listing all the certificate formats it knows and the applicable Host: header.
The server would respond with the certificate for that host-name and the browser would validate it. After that, the HTTP connection would switch into TLS mode using the key in that certificate.
If the server responds to this initial request with 404 or some other error, the browser either shows an error or continues in insecure mode.