How fast does $|D_k\cap[1,n]|$ grow for the factorial-product-square index $F(m)=k$? (Erdős #374)

For $m\in\mathbb{N}$ let $F(m)$ be the least integer $k\geq 2$ (if one exists) for which there are integers $a_1<a_2<\cdots<a_k=m$ with the product $a_1!\,a_2!\cdots a_k!$ a perfect square, and set $D_k=\{m : F(m)=k\}$. Determine the order of growth of $\lvert D_k\cap\{1,\ldots,n\}\rvert$ for each $3\leq k\leq 6$. In particular, is it true that $\lvert D_6\cap\{1,\ldots,n\}\rvert \gg n$ (that is, does $D_6$ have positive lower density)?

Acceptance. FULLY RESOLVES: determine, with proof, the order of growth of $\lvert D_k\cap\{1,\ldots,n\}\rvert$ (an asymptotic $\sim c_k n^{\theta_k}$, or matching upper and lower bounds) for every $3\leq k\leq 6$; in particular a proof settling whether $\lvert D_6\cap\{1,\ldots,n\}\rvert\gg n$. The density statements may be proved analytically or constructively (an explicit positive-density family placing infinitely many $m$ in a given $D_k$, verified by exhibiting the factorial subsets). ADVANCES (each independently checkable): a proof of a new upper or lower bound on $\lvert D_k\cap\{1,\ldots,n\}\rvert$ for some $3\leq k\leq 6$ that strictly improves on what is stated in the background; OR an explicit infinite family witnessing $\lvert D_k\cap\{1,\ldots,n\}\rvert \gg n^{\theta}$ for a stated $\theta$, each member accompanied by its certifying factorial subset; OR a reproducible exact computation of $F(m)$ for all $m\leq N$ at a new record $N$, extending OEIS A387184/A388851/A389117/A389148 and reporting the measured growth of each $\lvert D_k\cap\{1,\ldots,N\}\rvert$ with the source code and machine-checkable square-certificates. Deliver the proof, the explicit family with certificates, or the computation code plus the extended tables.