  For a fixed prime $p$, we consider a filtration of the commuting complex of
elements of order $p$ in the symmetric group $\mathfrak{S}_n$. The filtration
is obtained by imposing successively relaxed bounds on the number of disjoint
$p$-cycles in the cycle decomposition of the elements. We show that each term
in the filtration becomes highly acyclic as $n$ increases. We use
$\mathbf{FI}$-modules in the proof.
