Problem is: How would it do that? In order to for a machine to accurately decide whether its conversation partner is a human or another machine, it would require a model that already accurately understands the difference.
So it's basically a hen-and-egg problem; In order to make the Discriminator for a GAN that should pass the Turing test, I need a machine that could already pass the Turing Test.
Sure, we could feed it billions of human conversations, and train it to recognize some sort of difference, and that could result in a Generator which somewhat mimics the way human conversation works, but that's not what the Turing Test evaluates...when I ask something that is nonsensical like "what does the magnetopause smell like", and the system simply simulates a human speech pattern along the lines of "i don't know, I never tried smelling it", instead of showing actual understanding about the universe that tells it that the question is absurd, it may fool the Discriminator-Algorithm of the GAN, but its not going to convince humans for long.