We consider the problem of steady-state generation of one-dimensional Bose gas tubes that are weakly coupled to each other via density-density interactions. We analyze this development through the Boltzmann collision integral approach. We argue that when single-particle Hall excitations dominate the dominant orders of collision integrals created in individual gases, the state of the gas first evolves to a non-thermal fixed point, that is, a preheating plateau. This order is dominant when pairs of tubes are unequal, for example, at different temperatures or with different effective interaction parameters $\gamma$. We characterize this unheated preheating plateau and construct both a quasi-conserved quantity that controls the existence of this plateau and a related generalized Gibbs ensemble.