It probably cannot be, so it is not an argument either way. The shroud indisputably passed through many hands, was rescued from a fire, is certain to have been exposed, being an object of veneration, to centuries of burning incense.
What is indisputable is Palestinian pollen. That it has medieval admixtures is just consistent with its manner of storage.
annalex: "It probably cannot be, so it is not an argument either way."
I think current state-of-the-art carbon 14 dating technology could overcome most or all of the problems you mention. Indeed, in an earlier post Swordmaker himself suggested that unofficial carbon 14 tests had been done on tiny tiny samples, indicating a very early Shroud age.
So, the real problem is not technology, it's the Church not allowing more tests. This naturally raises suspicions that the Church might be afraid of results.
Suppose, for example, tests indicated the Shroud originated in ancient Israel, but not in the 1st century AD -- say in the 1st century BC, or the third century AD. Now we'd have a case of "close but no cigar."
And even if the Shroud were reliably dated, what "chain of custody" proof do we have it was in any way related to the death of Jesus?
So, far too many questions about the Shroud have not been, or can not possibly be, answered for anyone to claim it's authenticity is "proved," much less to justify name-calling and disparagement of those who doubt it.