In Scientific American (Sept 2014), Lee Billings writes:
Lloyd, though, readily admits the speculative nature of CTCs. “I have no idea which model is really right. Probably both of them are wrong,” he says. Of course, he adds, the other possibility is that Hawking is correct, “that CTCs simply don't and cannot exist." Time-travel party planners should save the champagne for themselves—their hoped-for future guests seem unlikely to arrive.
What authoritative physicists say closed timelike curves (CTCs) exist?
Further to a comment, here is an article about an experiment apparently using post-selection closed timelike curves, (P-CTCs)
Because P-CTCs are based on post-selected teleportation, their predictions can be experimentally demonstrated. To experimentally demonstrate the grandfather paradox, we store two qubits in a single photon: one in the polarization degree of freedom, which represents the forward-travelling qubit, and one in a path degree of freedom representing the backward travelling qubit as shown in Fig 3.
... probe qubits measure the state of the polarization qubit before and after the quantum gun is “fired”. When the post-selection succeeds (i.e. the time travel occurs), the state of the probe qubits is measured.