Has that been established? I know one of the critical issues is obtaining samples for analysis and the SOT Church officials are very, very reluctant to allow a lot of sampling.
It is not just the Cathedral in Turin. The Shroud itself is considered the personal property of the current Pope, who ever that may be. . . and the Pope has to sign off on any thing done to or on it. The Last three popes have only allowed the ill-advised "restoration" of the Shroud in 2002, which did far more damage to the cloth than any preservation. The restoration probably also did more damage to the DNA that is present.
All the blood and sera that has been examined so far has shown very damaged DNA with broken, fragmented strands.
The status of DNA studies from the FAQ on Shroud.com says:
Q: Has DNA testing ever been performed on the Shroud, assuming the blood stains still have traces of DNA?A: Several years ago, some Texas researchers did a DNA study of supposed Shroud bloodstains, but the provenance of the samples they used was questionable and their results have not been officially recognized. Nonetheless, their findings concluded that the blood on the Shroud is from a male human. They also stated that the blood is so old and degraded that very few DNA segments were found, eliminating any possibility of "cloning" anything from the blood found on the cloth. Other DNA experts argue however, that so much contamination exists on the Shroud that no DNA test, no matter how carefully done, could ever be considered definitive. During the 1978 exhibition and scientific examination, the cloth was handled by many people, including most members of STURP, the Church authorities who prepared it for display, the Poor Clare nuns who unstitched portions of it, visiting dignitaries (including the Archbishop of Turin and the emissary of King Umberto) and countless others. During the five days and nights of the 1978 examination, the Shroud was continuously exposed to contamination as it lay unprotected on the support table. Every member of our research team, including myself, left DNA on the cloth. And remember, the cloth has been displayed and handled thousands of times throughout its history. Once again, the Shroud presents us with an enigma that even DNA evidence may not definitively unravel.
On the other hand, DNA evidence does little if anything to help determine the mechanism that formed the image on the cloth. I am not sure that it has much other value, except perhaps, to satisfy someone's curiosity. I personally see little merit in pursuing it and expect the Church will not allow any formal DNA testing in the forseeable future anyway. The authorities have already officially stated that any future research efforts will concentrate exclusively on the preservation and conservation of the Shroud.