Erdős problem 390

Let $f(n)$ be the least $m$ for which $n!$ can be written as $a_1\cdots a_k$ with $n < a_1 < \cdots < a_k = m$ - the smallest possible largest factor in a factorization of $n!$ into distinct integers all exceeding $n$. Erdős, Guy and Selfridge proved $f(n) - 2n \asymp n/\log n$. Erdős asked whether there is a constant $c$ with $$f(n) - 2n \sim c\,\frac{n}{\log n},$$ and what it is.