With a sprinkling of legends. Probably as authentic as the "James" reliquary.
That 4th Century part smacks of the purveyors of pieces of the "True Cross".
Mind you, I'm not implying that the church authorities were involved in a fraud, but rather perhaps they were duped into buying the relics from a con man. Of course, after the passage of 17 centuries, they now really are, rather ironically, relics!
1,000 years ago nobody would dare to sell any such thing to the Church. Things were given freely, and passed from one generation of priests to another. That's at least what I gather from reading stories of Brother Cadfael.
Besides, 1,000 or 2,000 years ago people may have been not very informed about quantum mechanics, but they were just as good at reading other people's actions and intentions as we are today. Perhaps they were even better at that, given that *all* interactions at that time were face to face. You just can't become an Abbot or other Church leader without good "people skills."
Something tells me that it wouldn't be trivial for a medieval con man to even *give* the fake relics to the Church. That time was much closer to the actual events than we are now, and the Church would be in possession of records that are now lost to us. They also took religion seriously, and wouldn't commit a sin by honoring unvetted relics. They also had lots of monks eager to hit the road, retrace the trail of relics and get confirmation of every claimed fact.
Note that the Shroud of Turin is still not accepted by the Church as genuine, just because there is no certainty in its identification. The Church typically errs on the side of caution in cases like that.