Why do requests to Websocket's $connect route doesn't fail on "400" error code from http integration?

In my application, I integrate $connect the route with my HTTP endpoint using HTTP integration and 'Content Handling' = 'passthrough'. There is an example in the documentation that says that if I return 400 status from lambda integration, the connection request fails: https://docs.amazonaws.cn/en_us/apigateway/latest/developerguide/websocket-connect-route-subprotocol.html

However, no matter what code I return from my HTTP integration endpoint, the connection is always successful. The only way how I can fail my connection is to do the checks in the authorizer and fail it from there, but this is very inconvenient in my case because my authorizer must be a part of my HTTP endpoint logic instead of extracted to the separate authorizer.

Can I fail the connection attempt from an HTTP integration attached to the $connect route?