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The moral of the story is that frequentism and Science do not mix. Let me say it directly: you should be suspicious of the use of frequentist confidence intervals and p-values in science. In a scientific setting, confidence intervals, and closely-related p-values, provide the correct answer to the wrong question. In particular, if you ever find someone stating or implying that a 95% confidence interval is 95% certain to contain a parameter of interest, do not trust their interpretation or their results. If you happen to be peer-reviewing the paper, reject it. Their data do not back-up their conclusion.
</spoiler>
https://jakevdp.github.io/blog/2014/06/12/frequentism-and-ba...
However I would not expect CERN papers to make the kinds of terminological / theoretical lapses about confidence the parent thread was talking about. Papers should be rejected for that kind of error, even if you are not a bayesian.
In the case that they differ you almost always find that you have very few observations. I would argue that this 'difference' is not that exciting because it must be dominated by your assumptions, not your observations. After all once you accumulate enough observations the two methods tend to converge.
Personal conclusion: if the methods disagree work on getting more data instead of fighting over which method is better.