Ergodic many-body systems are expected to reach quasi-thermal equilibrium. Here we demonstrate that, surprisingly, high-energy electrons, which are injected into an interacting one-dimensional quantum Hall edge mode, stabilize at a far-from-thermalized state over a long-time scale. To detect this non-equilibrium state, one positions an energy-resolved detector downstream of the point of injection.  While in the case of short-ranged interactions in multi-channel systems the energy distribution generically relaxes to a near-thermal asymptotic state, we here consider a screened interaction of finite range and obtain a different result:  the resulting many-body state comprises fast-decaying transient components, followed by a nearly stationary distribution with a peak near the injection energy.