Does the Church hold that this is in fact the garment worn by Christ, or does it not??
They THINK it is but, of course, no one knows for sure, that is, the Church does NOT hold to it as a FACT.
There have been TONS of fabulous fakes throughout the millenia of Christendom and Catholics are only required to believe in dogma, NOT in every relic, true or not, that surfaces.
I suppose it could possibly be done on something like a Navajo rug loom, but the advantage of double-layer weaving an everyday garment like that, as opposed to sewing the front and back together, escapes me.
No, the Church does not hold that this is the garment of Our Lord. It has been around for a long time, has a documented history and shares some aspects of the Shroud of Turin. With that in mind the Church will display it to the faithful from time to time without making a definitive statement because she cannot. One is not compelled to believe anything about it.