The Soviets probably had nothing to do with the hoax either.
If the Bay of Pigs invasion was a hoax, as I suspect it was, the Soviets would have known, Castro would have known, and the Pentagon would have known. This leads me to suspect that, given that everyone knew everything, that the entire Cuban Missile Crisis was a hoax. By “hoax” I mean no showdown, no Soviet vessels turning around, no blockade, no missiles. It was all a TV show.