That is, until one brave man stepped forward.
The ENEA has determined that the imprint on the shroud comes from an intense flash of light. A local professor of chemistry, Luigi Garlaschelli, has determined that such a flash of light, because no such technology is known to exist at the time that could produce such brilliant light, must have supernatural origins.
Is he right, or is he wrong?
Well, we certainly can't say with 100% certainty he's wrong - after all, you can't prove a negative, but merely prove other explanations to be so likely as to render alternative explanations so unlikely as to be treated as not a possible explanation. Can we do that here? Well, let's examine possible explanations of things that could produce a burst of UV rays powerful enough to burn the image of a man into a shroud.
Supposing that aliens visited the planet at the time of the shroud's construction, they would probably have sufficient technology to generate such UV rays if they had mastered interstellar and/or intergalactic travel. However, there's no evidence to point to aliens visiting earth prior to the 20th century, during which alien visitation stories took the world by storm, so this seems unlikely. Incidentally, I recommend everyone read Carl Sagan's "The Demon-Haunted World", where he expounds on the epidemic of alien visitation stories that cropped (no pun intended) up.
Jesus Christ (it's a lion!)
Maybe it really was Jesus - he was killed and wrapped in this shroud, and his holy brilliance burned his likeness into the cloth. Ignoring the lack of historical evidence for the Jesus described by the Bible, this explanation faces the problem of the carbon dating placing its creation no earlier than 1260 CE, which is roughly 1,230 years later than the alleged lifetime of Jesus Christ.
We could suppose that the carbon-dating got it wrong - that, in fact, the carbon-dating process is mistaken (as so many young-Earth creationists pine for), or that maybe the shroud was patched and the fabric they tested was much later than the rest of the shroud. This, however, seems unlikely, as nothing of studies, including this most-recent one, indicates as such.
This one seems to have the strongest evidence, as there's no record of a visitation by Jesus Christ involving this shroud in the 13th to 14th century. However, this doesn't answer the big question: is Luigi Garlaschelli Wrong?
My lack of scientific expertise aside, I can't say 100% that Garlaschelli's allegation of supernatural origins is wrong due to any impossibility of the supernatural. I feel confident, however, in saying that he's wrongly come to his conclusion. What we have here is a conflict of explanations: carbon-dating that places the shroud in the 13th to 14th century (and such dating remains unchallenged), and an explanation of the imprint that would appear to require technology not known to exist in the 13th or 14th century. What Garlaschelli should have said - and, indeed, what any scientist should say - is, "Huh. That's a very interesting/weird contradiction. We should validate the dating and the means by which the ENEA determined the means of producing the imprint to make sure both are accurate conclusions. If they both turn out to still be accurate, perhaps we should investigate other means by which the shroud may have been imprinted with the image of a man."
Thus, Garlaschelli's explanation of "supernatural origins" is wrong - not because it's not possible for the imprint to have supernatural origins, but because he's made a claim with certainty without supporting evidence. In the end, he wants the shroud to have supernatural origins, and is thus willing to fill in the "I don't know gap" left behind by the contradiction of the carbon-dating and UV burst origins of the imprint with his own desire to prove the existence of his god.