OPEN

Let $A(n)$ denote the least value of $t$ such that
\[n!=a_1\cdots a_t\]
with $a_1\leq \cdots \leq a_t\leq n^2$. Is it true that
\[A(n)=\frac{n}{2}-\frac{n}{2\log n}+o\left(\frac{n}{\log n}\right)?\]

If we change the condition to $a_t\leq n$ it can be shown that
\[A(n)=n-\frac{n}{\log n}+o\left(\frac{n}{\log n}\right)\]
via a greedy decomposition (use $n$ as often as possible, then $n-1$, and so on). Other questions can be asked for other restrictions on the sizes of the $a_t$.