The Shroud is now immersed in a bath of some chemical to protect it from decaying more. This chemical bath evidently prohibits any accurate carbon testing now. It really is not important. The Shroud has been proven authentic...the question is not is it real or fake, but is it Jesus or not. There is not way it was created in the middle ages as the technology for knowing the different light wave frequencies that have been used to prove it’s authenticity were known back then...the image is also contains 3 dimensional information...that precludes it from being any kind of painting, dyeing, etc.
I disagree that it's been "proven authentic". It has not been falsified is the best we can say right now. No one can prove it's a fraud, forgery, fake, or a work of art. We don't know.
The evidence leans heavily toward it being of first century provenance and that at sometime in it's existence it was in an area in a 50 mile radius around Jerusalem, had plants that were native to that area placed on it, and had the portion of it that was the backside of the dorsal image (and on the front of the cloth under the feet of the image) laid on a surface that had stone dust from Travertine Aragonite that is unique to quarries to the east of Jerusalem and consistent with that found in the traditional sites associated with Christ's tomb.