What is the size of the largest $A\subseteq \{1,\ldots,n\}$ such that if $a\leq b\leq c\leq d\in A$ are such that $abcd$ is a square then $ad=bc$?

Cambie and Weisenberg have noted in the comments that the set of semiprimes also works, showing $(1+o(1))\frac{\log\log n}{\log n}n \leq \lvert A\rvert$ is achievable.