Perhaps one way to assess the debate is to ask, is modern cosmology approaching "closure" or not, where I define "closure" to mean, the point where new independent observations agree with the current deduced parameter set-- they do not require the introduction of new parameters. It was my impression that modern cosmology is indeed approaching closure-- it's true that we got three big kicks in the pants with the need for inflation, dark matter, and dark energy, but more recent observations have not required new interpretations or new parameters. The convergence of the Hubble parameter is kind of a subset of this overall trend, and the other key parameters seem to be coming into focus more and more. For me to think cosmology is really in such a sorry state, I would need to see some new independent cosmological measurement that also requires the introduction of a new free parameter into the model. If I see that, then I'll say "OK, Disney was right", but if new observations just require minor tinkering, I'll conclude the opposite. Perhaps the jury is still out on that.
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