SOLVED
Is it true that if $A\subseteq\mathbb{N}$ is such that
\[\frac{1}{\log\log x}\sum_{n\in A\cap [1,x)}\frac{1}{n}\to \infty\]
then
\[\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\to \infty?\]
Tao
[Ta24b] has shown this is false: there exists $A\subset\mathbb{N}$ such that
\[\sum_{n\in A\cap [1,x)}\frac{1}{n}\gg \exp((\tfrac{1}{2}+o(1))\sqrt{\log\log x}\log\log\log x)\]
and
\[\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\ll 1.\]
Moreover, Tao shows this is the best possible result, in that if $\sum_{n\in A\cap [1,x)}\frac{1}{n}$ grows faster than $\exp(O(\sqrt{\log\log x}\log\log\log x))$ then
\[\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\to \infty.\]