The Vatican Museum states that the man to the right of Marcantonio Raimondi, who others have thought is Durer - is Raphael. And that does look like the Veil man. I assume the description means the viewer’s right and not the man in the picture’s right.
Apparently, the view that it is Raphael is a fairly new view as remarked by one author as being a 20th century tradition. Not sure when this idea first came up. Raphael’s patron, Julius II , is the one being carried.
Julius II would obviously have access to byssus as we use the term today.
Now I see what the writer meant about the 20th century view.
Vasari writes that some think it is Guilo Romano in the fresco and others say it is Baldassare Peruzzi, an architect of Saint Peter’s basilica. So who is the mystery man who looks like the man in the Veil.....