We briefly review an important shortcoming-unearthed in previous works-of the standard version of the inflationary model for the emergence of the seeds of cosmic structure. We consider here some consequences emerging from a proposal inspired on ideas of Penrose and Diósi about a quantum-gravity induced reduction of the wave function, which has been put forward to address the shortcomings, arguing that its effect on the inflaton field is what can lead to the emergence of the seeds of cosmic structure. The proposal leads to a deviation of the primordial spectrum from the scale-invariant Harrison-Zel'dovich one, and consequently, to a different CMB power spectrum. We perform statistical analyses to test two quantum collapse schemes with recent...