An extended Enskog asymptotic expansion method is used to find approximate solutions of the Boltzmann equation in the cases of strong thermal nonequilibrium. The formalism to determine thermal conductivities in the expanding flow of a binary mixture of a diatomic and its parent atoms through a supersonic nozzle is applied. Results show that the diffusion of highly excited vibrational levels, which become overpopulated during the expansion, substantially contributes to the transport of energy. This effect cannot be represented by simplified theories such as the multitemperature model.