The trivial lower bound for the girth of $S_n$

Consider the Cayley graph of $S_n$ generated by a random pair of elements $x,y$. Conjecturally, the girth of this graph is $\mathrm{\Omega }(n\log n)$ with probability tending to $1$ as $n\to \mathrm{\infty }$. We show that it is at least $\mathrm{\Omega }(n^{1\mathrm{/}3})$.