Gurzadyan and Penrose are saying that there is observational evidence for a particular cosmological model known as Conformal Cyclic Cosmology (CCC). Specifically, the evidence comes from measurements gathered by the Wilkinson Microwave Anisotropy Probe (WMAP) (a satellite) and Boomerang '98 (a balloon flight above Antarctica in 1998, which also measured temperature fluctuations in the cosmic microwave background).
Specifically, they claim to see the signatures of supermassive black hole encounters in the aeon prior to the Big Bang - these show up as families of concentric circles over which the temperature variance is anomalously low in the cosmic microwave background radiation.
Of course, other cosmological models might also be consistent with the measurements they cite.
So, back to my original hypothesis..
"...to reproduce the effects that we appear to see, within the framework of inflation, one would require a mechanism for producing recurrent explosive events close to the inflationary turn-off point. No such mechanism has ever been seriously contemplated."
(but someone might start seriously contemplating it now!)