The present work extends an earlier study of consequences following from a re-ordering procedure applied to the collision integrals in the coupled Boltzmann equations for a binary gas. The re-ordering is that originally suggested by Grad for describing disparate-mass mixtures, and allows independent species temperatures and flow velocities. First order expressions for the heat flux and pressure tensor are obtained, and some evaluation procedures are discussed. It is shown that the approach leads to a physically unreasonable prediction for the coefficient of thermal conductivity, whereas the classical Chapman-Enskog prediction agrees with a simple mean-free path-estimate. This negative conclusion is shown to be unaffected by second-order corrections.