The 13th century result has been proven erroneous due to a repair known as "French reserving."
Unfortunately, the preservative medium in which the Shroud has been stored since the original carbon dating precludes an accurate result were the Shroud tested again from a pristine sample.
Make that "French Invisible Reweaving" and you'd be right. It was a technique developed in the 16th Century to repair tapestries and arrases that had frayed or been moth eaten so that the original images woven into them would not show the patch by "reweaving" new threads invisibly into the old, pre-dyed and matching the old ones. It required very skilled artisans to do the work.