Square products of factorials and a conjecture of Erdős and Graham

For $n\ge2$ let $F(n)$ be the least $k\ge2$ such that $n!$ is the largest factor in a product of $k$ distinct factorials that is a perfect square, and let $D_k(X)$ be the number of $n\le X$ with $F(n)=k$. Erdos and Graham asked for the order of growth of $D_k(X)$ for $3\le k\le6$, and conjectured th