That painting is based on the Shroud of Turin image of Christ, which is represented in my avatar, and is the most accurate one......
No, afraid not. The icon of Christ Pantocrator from Sinai predates the first appearance of the Shroud of Turin in history by centuries. Now, what you say may, perhaps, be right in a sort of roundabout way in that there is a suggestion that iconographical depiction of Christ as bearded and with long hair are due to the rediscovery of the Mandylion of Edessa in the 6th century (though that was only ever described as a square with the image of Christ's face on it) and that there has likewise been some suggestion that the Shroud of Turin might be the Mandylion as that disappeared from Constantinople during the looting of the 4th Crusade and the Shroud of Turin first appeared in the west over 150 years later.
I think that, to be honest, there are just too many holes in the story, though. The Mandylion isn't a burial shroud, only has a facial image and is alwways described as square - the Shroud of Turin just isn't like that and why would the Crusaders wait so long before unveiling their prize if they'd taken the Shroud from Constantinople? They weren't shy about installing the rest of their pilfered relics in western churches. For comparison, the icon from St. Catherine's monastery dates from 6th century.
James