Of course, there's no way the Shroud's authenticity can be "proved," since even if carbon-14 tests showed it to be 1st century, how does that make the figure Jesus?
On the other hand, it could be "disproved," if reliable tests showed some other time period. Are such tests even possible, given all the circumstances? I don't know, perhaps not today.
But I'd suppose that the challenge of finding a reliable method for testing the Shroud would work on the imaginations of various lab operators. And in due time, one or more will come up with a set of answers that could succeed.
Until that day, the Church has every right to insist on "no more nonsense." After that day, it will need to ask itself does it really want to risk seeing the Shroud disproved?
The Church does not consider the shroud “proved”. One is not required to believe anything in particular about the shroud. There are signs pointing to that being authentic shroud of Christ, — its unexplained physical properties, consistence with what we do know of Christ and 1c Palestine, and the legent around it.
For nearly sure it is a burial cloth of someone, so it deserves respect on that level. It does not deserve to be ripped to shreds out of some curiosity.
It surely reminds us powerfully of Christ. I am convinced that even if disproved by carbon dating or something else it will retain that power.