[simmervigor]

The vanilla CONNECT method is an instruction to open a TCP connection to the target server. What is sent over that is entirely up to client and target the server it doesn't need to match. Its often TLS carrying HTTP but it could be anything.

The proxy could inspect the traffic it is carrying and try to enforce some access control or policy. However, the use of TLS or other encrypted protocols limits the ability to do see what is happening. Then you get into a different logical layer of whether there is MITM happening, but that is tangential to conventional use case for HTTP proxy.