Local newforms for generic representations of $p$-adic ${\mathrm{S}\mathrm{O}}_{2n+1}$: Reduction

We prove that if the space of newforms is non-zero for every irreducible generic supercuspidal representation of ${\mathrm{S}\mathrm{O}}_{2n+1}$ then it is also non-zero for all irreducible generic representations of ${\mathrm{S}\mathrm{O}}_{2n+1}$.