There's different senses of the word information, so maybe a cosmologist would hesitate answering this until you explained exactly what you meant. In objective collapse models, it is just not conserved because collapses of the wave function don't conserve information. I suspect though that what Neil has in mind is not micro physical information that gives us the exact micro state of the universe but something more like accessible macroscopic information. Such information can be lost and gained depending on different physical processes.

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