Its interesting that Matthew, Mark and Luke all mention that the Hemorrhissa had suffered for 12yrs of her plague. Whilst this may be a common source indicator would that period of time have any other symbolism?
The story of Veronica or Bernice seems to be a folk tale but isnt mentioned in the Synoptic Gospels. In 1143 the "Sudarium Christi which is called Veronica" was being 'censed' in Rome
However the Middle Ages were awash with holy relics or acheiropoietos similar in design or appearence. Although a cautionary note ascribes this word to a genre of Religious art rather than images 'not made by human hands'.
There were Sudariums of Andechs in Bavaria and Oviedo in Spain.
The Lirey shroud once in the posession of de Charny was allegedly the Shroud stolen from Besancon Cathedral and substituted with a copy.
The Cistercians at Cadouin in the Perigord had what they believed to be The Sudarium until it was discovered in 1933 to be 10cent Egyptian and covered with Koranic inscriptions. So, naturally, it was destroyed.
If the Turin Shroud's history is reliably charted from 1353 to present day then its earlier history is less so. Wilsons support for the Mandylion notion has the advantage of it being walled up and forgotten in Edessa for 500yrs. A handy way to jump half a milenium after its arrival there with Thaddeus.
Its arrival in Constantinople and being seen by de Clari as the 'Sydoine' in 1203 before the sacking in 1204 does seem to place it at the scene, and where else would an object of such repute been in 12/13th century?
So, if you discount the Knight/Lomas theory of de Molays shroud or Laidlers contention that a proto photograph image created it you are left with others like Currer-Biggs and Wilson who see the Mandylion and Shroud as one and the same.
Fascinating subject the shroud, even without entering the twilight world of Prince and Pickett.