No: ESP would confound the test
Extrasensory Perception could invalidate the Turing Test in a variety of ways—eg if a competitor with ESP could "listen in" on the judges & gain an unfair advantage, or a judge with ESP could easily discern humans from machines by clairvoyance.
Argument anticipated by Alan Turing, 1950.
As Turing observed:
"with ESP anything may happen".
The Turing Argument"A more specific argument based on ESP might run as follows: 'Let us play the imitation game, using as witnesses a man who is good as a telepathic receiver, and a digital computer. The interrogator can ask such questions as 'What suit does the card in my right hand belong to?' The man by telepathy or clairvoyance gives the right answer 130 times out of 400 cards. The machine can only guess at random, and perhaps gets 104 right, so the interrogator makes the right identification.' There is an interesting possibility which opens here. Suppose the digital computer contains a random number generator. Then it will be natural to use this to decide what answer to give. But then the random number generator will be subject to the psychokinetic powers of the interrogator. Perhaps this psychokinesis might cause the machine to guess right more often than would be expected on a probability calculation, so that the interrogator might still be unable to make the right identification. On the other hand, he might be able to guess right without any questioning, by clairvoyance. With ESP anything may happen"Turing's Comments about ESP"This argument is to my mind quite a strong one. One can say in reply that many scientific theories seem to remain workable in practice, in spite of clashing with ESP; that in fact one can get along very nicely if one forgets about it. This rather cold comfort, and one fears that thinking is just the kind of phenonmenom where ESP may be especially relevant." (Turing, 1950)
Source: Turing, A.M. (1950) "Computing Machinery and Intelligence" Mind, 59, 433-60.