Erdős Problem #844 — Largest Sets with No Squarefree Pairwise Products

For $N\in\mathbb N$, let $E_N=\{n\in\{1,\ldots,N\}:2\mid n\text{ or }n\text{ is not squarefree}\}$. Is it true that, for every $N$, the maximum cardinality of a finite set $A\subseteq\{1,\ldots,N\}$ satisfying

$$\forall a,b\in A,\quad ab\text{ is not squarefree}$$

is $|E_N|$?