In Theorem 7.6 of Illusie - An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology it is stated that if $X$ is a log regular fs log scheme and $U$ is the open subschema where the log structure is trivial, then the Kummer-étale site of $X$ is equivalent to the curve-tame site of $U$. The proof is not written there, and it cites a mysterious paper of Fujiwara and Kato that apparently was never published (or at least I have not been able to find it). This result seems to be well known and (as far as I understand) not trivial at all, but I cannot find a reference where this is proved.
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He did not say that the sites are equal as this claim is wrong in general, but that the categories of covers, i.e. finite morphisms which cover the log scheme (See definition 3.1. of the same paper), are equivalent. There is a written proof in theorem 13.3.45 of Almost purity and perfectoid spaces by Gabber and Romero.
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$\begingroup$ I am having trouble inferring the end of the sentence: "but the category of covers …". $\endgroup$– LSpiceCommented Nov 15, 2022 at 0:48
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1$\begingroup$ You are right. I finished the sentence. $\endgroup$ Commented Nov 15, 2022 at 1:03