We theoretically investigate the nonequilibrium current through a quantum dot coupled to one-dimensional electron leads, utilizing a controlled frequency-dependent renormalization group approach. We compute the nonequilibrium conductance for large bias voltages and address the interplay between decoherence, Kondo entanglement, and Luttinger physics. The combined effect of large bias voltage and strong interactions in the leads, known to stabilize two-channel Kondo physics, results in nontrivial modifications in the conductance. Interestingly, these unusual changes in the conductance persist in the presence of a finite channel asymmetry.