Symmetric chain decomposition of necklace posets

A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $P$ is any symmetric chain order, we prove that $P^n\mathrm{/}{\mathbb{Z}}_n$ is also a symmetric chain order, where ${\mathbb{Z}}_n$ acts on $P^n$ by cyclic permutation of the factors.