Disproved. Tao's Theorem 1 (Integers 24 (2024), paper A100; arXiv:2407.04226, v5) constructs, for every $C_0>0$, a set $A$ with $\sum_{n\in A,\,n\le x}1/n=\exp((C_0/2+o(1))(\log\log x)^{1/2}\log\log\log x)$ and $\sum_{n,m\in A;\,n,m\le x}1/\mathrm{lcm}(n,m)\ll_{C_0}(\sum_{n\in A,\,n\le x}1/n)^2$; the first quantity grows faster than $\log\log x$, so the hypothesis holds and the conclusion fails. The paper's introduction (p. 4) also records the elementary counterexample of the squarefree numbers with exactly $k\ge2$ prime factors, implicit in earlier work of Bergelson and Richter. Theorem 1's growth rate is optimal up to $C_0$, which the site's commentary records as the best possible result. The paper is published in a refereed journal; arXiv v5 postdates the journal version and was not compared with it. The site's label is DISPROVED (LEAN); the suffix is a catalog label explained under Formalization, and no local kernel credit is claimed. The claim page is Tao's theorem (accepted; refereed, and credited by the site's curator for its label).