In this paper we resolve a controversy related to the demixing transition in the nematic phase of binary hard rod mixtures. Previous analyses do not agree on the existence of a consolute point closing the nematic-nematic coexistence region. We definitely rule out the existence of this critical point, at least in the Onsager theory. Our analysis requires the determination of self-consistent orientation distribution functions, for which we develop an efficient numerical scheme.; This scheme is based on the scaling properties of the high density limit of the stationarity condition on the free energy functional. We illustrate this method in some detail for the monodisperse thin hard rod system.