We prove the convergence of Araki-Haag detectors in Haag-Kastler quantum field theory with an upper and lower mass gap. We cover the case of a single Araki-Haag detector on states of bounded energy, which are selected from the absolutely continuous part of the energy-momentum spectrum below the three-particle threshold. In our proof, we apply Mourre's conjugate operator method for the first time in Haag-Kastler quantum field theory. Taking quantum mechanical proofs of asymptotic completeness as a guide, these advances bring us closer to understanding two-particle asymptotic completeness in local relativistic quantum field theory.