I would suggest that this is not “plausible”:
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It appears that the first version of the tradition, that of Eusebius, does not mention an image.
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The first version to mention an image, the Doctrine of Addai, describes it as a portrait of Jesus painted by a court painter for King Abgar of Edessa.
When Hannan, the keeper of the archives, saw that Jesus spake thus to him, by virtue of being the king’s painter, he took and painted a likeness of Jesus with choice paints, and brought with him to Abgar the king, his master.[1]
So what we are left with is a highly-mutable legend that makes no mention of a burial shroud.
This is NOT “follow[ing] the evidence wherever it lead[s]”.
This is manipulating the evidence to try to force it to fit a predetermined narrative!
Or, in the words of The X-files’ Fox Mulder: