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Under the CMB method, it sounds like the calculation gives the same expansion rate everywhere. Under the Cepheid method, they get a different expansion rate, but it’s the same in every direction. Apparently, this isn’t the first time it’s been seen. What’s new here is that they did the calculation for 1000 Cepheid variable stars. So, they’ve confirmed an already known discrepancy isn’t down to something weird on the few they’ve looked at in the past.
So, the conflict here is likely down to our understanding of ether the CMB or Cepheid variables.
I wish the article had broken it down the way you did. Thanks!