Outgoing proxy connection problem

Not sure if this helps, but you can try..

So WordPress’ default HTTP transport is using cURL, and the HTTP class sets the proxy authorization method to CURLAUTH_ANY (see source), which might explain why the Proxy-Authorization: Basic xxxxx... header is missing from the request.

And there’s a hook you can try to change the proxy authorization method to CURLAUTH_BASIC (which is widely supported, but also very insecure), CURLAUTH_NTLM or one of the other available methods.

So add this to your theme functions.php file.. (or put the code into a Must-Use plugin):

add_action( 'http_api_curl', function ( $handle, $r, $url ) {
    $proxy = new WP_HTTP_Proxy();
    if ( $proxy->is_enabled() && $proxy->send_through_proxy( $url ) ) {
        curl_setopt( $handle, CURLOPT_PROXYAUTH, CURLAUTH_BASIC );
    }
}, 10, 3 );

Alternatively, you can set PHP Streams as the default HTTP transport…

add_filter( 'http_api_transports', function ( $transports ) {
    return [ 'streams', 'curl' ];
} );

See the hook’s reference and WP_Http_Streams for more information. But basically, the hook allows you to use a custom class, and WP_Http_Streams would always set the Proxy-Authorization: Basic xxxxx... header when necessary.