All anyone has to do is carbon date a couple of other sections of the shroud. Methinks no one really wants the answer here to come forth.
That may no longer be directly possible. It seems that after the "restoration" work done in 2002, the Shroud was placed into its casket after the wood of the casket had been immersed in Thymol... a high carbon content preservative... that may have irretrievably contaminated the Shroud.
However, there is a way to date the Shroud indirectly. During that same "restoration," the idiot who did it, Madame Bechtold Fluery-Lemburg, trimmed away some scorched areas of the Shroud that had been burned in the fire of 1535, under the mistaken opinion that the Shroud was still buring! Those trimmings have been stored and are not contaminated with Thymol. Certainly using those already excised pieces of the Shroud would do no further damage to the cloth.
Incidentally, an unauthorized C-14 test was done on a thread from the main body of the Shroud. It dated to the 1st Century, Plus or minus 50 years.