Hello Alessandro-
That is an interesting question. The GSS does not check the certificate chain or domain, thus, to directly answer your question, no, it doesn't care about trust. I have never tested if the clock is checked against the date range on the certificate, but I would imagine it doesn't check that either.
At this point, if you had a linux box, issue a wget or curl request against the server and see what it sends back for the the page your GSS is probing (GSS only accepts a 200ok.)
Regards,
Chris