And Tutankhenamen has a clearly 20th Century history... with very few mentions in history...
Yes, the Bishop did claim that. However, the letter only exists as a draft. No copy has been found in the Vatican archives (which for this period are very complete) and the evidence is that the Bishop Henri of Trois did not send it (it's possible someone removed it). Although the dispute did eventually come to the Pope's attention, the Pope allowed the continued exhibition (but not the claim that it was Christ's shroud) but also put Henri under a perpetual order of silence... he could never speak of the subject again.
It should also be noted that Henri may have had an ulterior economic motive... the close proximity of Lirey to Trois was resulting in a reduction in the donations to see his relics at his cathedral...
This is a really bad analogy, and it's not true anyway. There are inscriptions from his period, and the insinuation that the bishop was lying reflects a colossal will to believe in something with no history prior to that bishop's lifetime.