The complexity of the Lie module

We show that the complexity of the Lie module $\mathrm{L}\mathrm{i}\mathrm{e}(n)$ in characteristic $p$ is bounded above by $m$ where $p^m$ is the largest $p$-power dividing $n$ and, if $n$ is not a $p$-power, is equal to the maximum of the complexities of $\text{Lie}(p^i)$ for $1\le i\le m$.