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The functional renormalization group (FRG) provides a flexible tool to study correlations in low-dimensional electronic systems. In this paper, we present a novel FRG approach to the steady-state of quantum wires out of thermal equilibrium. Our method is correct up to second order in the two-particle interaction and accounts for inelastic scattering. We combine semi-analytic solutions of the flow equations with MPI parallelization techniques, which allows us to treat systems of up to 60 lattice sites. The equilibrium limit is well-understood and serves as a benchmark. We compute effective distribution functions, the local density of states, and the steady-state current and demonstrate that all of these quantities depend strongly on the choice of the cutoff employed within the FRG. Non-equilibrium is plagued by the lack of physical arguments in favor of a certain cutoff as well as by the appearance of secular higher-order terms which are only partly included in our approach. This demonstrates the inadequacy of a straightforward second-order FRG scheme to study interacting quantum wires out of equilibrium in the absence of a natural cutoff choice.