modus tollens wrong conclusions

If it is the Holy Shroud (p) then it is ancient (q)
The shroud under examination is not ancient (not q)
So there is no Holy Shroud (not p)

Looks like modus tollens, but it is wrong, since the fact that one shroud is not ancient does not imply that there are no ancient shrouds, one of which could be the Holy one. Is this a "faulty generalization"?
Thanks.

I'm not seeing the Modus Tollendo - only the non sequitur.

Non Sequitur (also known as: derailment, “that does not follow”, irrelevant reason, invalid inference, non-support, argument by scenario [form of], false premise [form of], questionable premise [form of], non-sequitur)

Description: When the conclusion does not follow from the premises. In more informal reasoning, it can be when what is presented as evidence or reason is irrelevant or adds very little support to the conclusion.

The premise is faulty because it assumes there is only one holy shroud by stating it is "the holy shroud".
There could be many holy shrouds from many various faiths - both ancient and neoteric.
Now, If it was referring specifically to The Shroud of Turin that might be another claim.

But then we'd have to agree on what constitutes ancient, as well as what exactly is holy.

The Shroud of Turin has been determined to be a forgery from the Middle Ages. So, it's possible it is neither ancient nor
holy. That is if one dismisses frauds and hoaxes as unholy and that the Middle Ages are not ancient enough.
It's possible it could be an antique, but not ancient.

There are just too many factors here that do not follow logically.

To simplify:

If it is the Holy Shroud (p) then it is ancient (q)
The shroud under examination is not ancient (not q)
So there is no Holy Shroud (not p) {WRONG!}

If it is the Holy Shroud (p) then it is ancient (q)
The shroud under examination is not ancient (not q)
So, the shroud under examination is not the Holy Shroud {CORRECT!}

Thus, this syllogism begs the question whether the Holy Shroud exists.

Much better!