Of the five knowledgeable people who initially thought that the fragment examined by Jull was not from the Shroud of Turin because it did not seem to be a 3-over-1 twill weave, four have gotten back to me. Those four people examined the problem further. All four of them have changed their minds and now concur that it is a 3-over-1 twill weave.
It just didn’t look like a 3-over-1 twill from the reverse side. Giulio Fanti sent me the following. It is a photograph of a piece of linen made to be exactly like the Shroud of Turin’s fabric pattern. On the left, the top is clearly 3-over-1. The bottom left shows how much the backside looks like a 2-over-1. It is clear how someone might have become confused. The right side of the image uses backlighting to reinforce how these patterns look:
This does not vindicate the Jull paper. It merely gets us past one problem.