We cannot examine the keep-alive status of the http parser in
`http_connect`; it's too late and the critical information about whether
keep-alive is supported has been destroyed.

Per the documentation for `http_should_keep_alive`:

> If http_should_keep_alive() in the on_headers_complete or
> on_message_complete callback returns 0, then this should be
> the last message on the connection.

Query then and set the state.