I had my doubts, so I looked up the etymology of the word fake. The most common origins were something along the lines of:
A likely source is feague "to spruce up by artificial means," from Ger. fegen "polish, sweep," also "to clear out, plunder" in colloquial use. "Much of our early thieves' slang is Ger. or Du., and dates from the Thirty Years' War" [Weekley]. Or it may be from L. facere "to do." Link
Feaguing does have a second meaning which is described in the post, but I couldn’t find mentions of that meaning linked to the word fake. But the practice of feaguing has apparently evolved from horses to humans as an act known as figging…I’ll leave to look that one up on your own (or not).