Covering numbers for characters of symmetric groups

If $n>4$ and $c(\theta )$ denotes the set of irreducible constituents of a character $\theta$, then $c({\chi }^k)=\mathrm{I}\mathrm{r}\mathrm{r}(S_n)$ for all nonlinear $\chi \in \mathrm{I}\mathrm{r}\mathrm{r}(S_n)$ if and only if $k\ge n-1$.