Summary: Let \(F\) be a non-Archimedean locally compact field of residual characteristic \(p\). Let \(\sigma\) be an irreducible smooth representation of the absolute Weil group \(\mathcal{W}_F\) of \(F\) and \(\operatorname{sw}(\sigma)\) the Swan exponent of \(\sigma\). Assume \(\operatorname{sw}(\sigma) \geqslant 1\). Let \(\mathcal{I}_F\) be the inertia subgroup of \(\mathcal{W}_F\) and \(\mathcal{P}_F\) the wild inertia subgroup. There is an essentially unique, finite, cyclic group \(\varSigma\), of order prime to \(p\), such that \(\sigma(\mathcal{I}_F) = \varSigma\sigma(\mathcal{P}_F)\). In response to a query of Mark Reeder, we show that the multiplicity in \(\sigma\) of any character of \(\varSigma\) is bounded by \(\operatorname{sw}(\sigma)\).