Section 6 of Computing Machinery and Intelligence works through the standard objections to thinking machines, then lands on the one Turing says is strongest. His words: these disturbing phenomena seem to deny all our usual scientific ideas, and unfortunately the statistical evidence, at least for telepathy, is overwhelming. He runs the scenario: a telepathic receiver names 130 of 400 cards where the machine manages 104, the interrogator spots the human, and the test fails. He adds that psychokinesis might corrupt the computer's random number generator. His fix is a building specification: put the competitors in a telepathy-proof room and the imitation game stands.
Textbooks reproduce the imitation game and omit the room. The story that survives in parapsychology circles is better than the one in computer science: the man who defined machine intelligence read the Society for Psychical Research seriously enough to firewall his test against it. The evidence he almost certainly meant was Samuel Soal's card-guessing sittings with Basil Shackleton, then the gold standard of telepathy research, conducted in the early 1940s and famous by 1950.
In 1978 Betty Marwick published a forensic reanalysis of Soal's raw score sheets and found patterns consistent with inserted hits: the flagship evidence of 1950 collapsed under suspicion of manipulation. The parapsychology establishment that supplied Turing's confidence spent the next decades unable to produce a replacement result of equal standing.
The evidence Turing called overwhelming was later shredded as manipulated. Did the test survive because he was wrong about telepathy, or because nobody ever built the room?