Tame multiplicity and conductor for local Galois representations

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 $\Cal W_F$ of $F$ and $\sw(\sigma)$ the Swan exponent of $\sigma$. Assume $\sw(\sigma) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and $\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\varSigma$, of order prime to $p$, so that $\sigma(\Cal I_F) = \sigma(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $\sigma$ of any character of $\varSigma$ is bounded by $\sw(\sigma)$.