A conjecture of Glasby, Praeger, and Unger on permutations of $S_n$

We prove a conjecture of Glasby, Praeger, and Unger concerning the symmetric group $S_n$. Let ${\pi }_n$ denote the proportion of elements of $S_n$ that are pre-$p$-cycles for some prime $p\in [2,n-3]$. We prove that ${\pi }_n>1\mathrm{/}3$ for all $n\ge 8$.