I don't think that they could possibly be confident that they've found relics from The Last Supper. How could they possibly identify them? And if you say "from the painting", I'll smack you, because the famous painting of it was made over 1000 years later :-P
The Shroud of Turin can't be proven real, or fake. So I will choose to believe that it's fake, because I'd rather be unsure than to have a false belief.