On product representations of squares

Fix $k \geq 2$. For any $N \geq 1$, 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. Erd\H{o}s, S\'ark\H{o}zy, and S\'os showed that $F_2(N) = (\frac{6}{\pi^2}+o(1)) N$, $F_3(N) = (1-o(1))N$, $F_k(N) \asymp N/\log N$ for even $k \geq 4$, and $F_k(N) \asymp N$ for odd $k \geq 5$. Erd\H{o}s then asked whether $F_k(N) = (1-o(1)) N$ for odd $k \geq 5$. Using a probabilistic argument, we answer this question in the negative.