Sondow's conjecture on the Smarandache function

For a positive integer $n$, let $S(n)$ be the smallest positive integer $j$ such that $n$ divides $j!$. Sondow's conjecture. The inequality

$$n^2<S(n)!$$

holds for almost all positive integers $n$. The paper proves the stronger assertion that, for every fixed real number $k>1$, $n^k<S(n)!$ holds for almost all positive integers $n$, so this conjecture is resolved.