On product representations of squares

Fix $k\ge 2$. For any $N\ge 1$, let $F_k(N)$ denote the cardinality of the largest subset of $\{1,\dots ,N\}$ that does not contain $k$ distinct elements whose product is a square. Erdős, Sárkőzy, and Sós showed that $F_2(N)=(\frac{6}{{\pi }^2}+o(1))N$, $F_3(N)=(1-o(1))N$, $F_k(N)\asymp N\mathrm{/}\log N$ for even $k\ge 4$, and $F_k(N)\asymp N$ for odd $k\ge 5$. Erdős then asked whether $F_k(N)=(1-o(1))N$ for odd $k\ge 5$. Using a probabilistic argument, we answer this question in the negative.