OPEN

Let $t(n)$ be the maximum $m$ such that
\[n!=a_1\cdots a_n\]
with $m=a_1\leq \cdots \leq a_n$. Obtain good upper bounds for $t(n)$. In particular does there exist some constant $c>0$ such that
\[t(n) \leq \frac{n}{e}-c\frac{n}{\log n}\]
for infinitely many $n$?

Erdős, Selfridge, and Straus have shown that
\[\lim \frac{t(n)}{n}=\frac{1}{e}.\]
Alladi and Grinstead [AlGr77] have obtained similar results when the $a_i$ are restricted to prime powers.