Proxy detection via "Proxy-Connection" header

I found a site that threw this message to me while trying to use Burp proxy:

I fired up wireshark and captured a normal request and a burp request, then exported the headers to a file

A quick diff showed the difference

Now it's just a matter of using burp "match and replace" feature

Finally we can load the page correctly, but what if instead of showing us that helpful message the application just behaved differently?