We consider a simple Maier-Saupe statistical model with the inclusion of disorder degrees of freedom to mimic the phase diagram of a mixture of rod-like and disc-like molecules. A quenched distribution of shapes leads to the existence of a stable biaxial nematic phase, in qualitative agreement with experimental findings for some ternary lyotropic liquid mixtures. An annealed distribution, however, which is more adequate to liquid mixtures, precludes the stability of this biaxial phase. We then use a two-temperature formalism, and assume a separation of relaxation times, to show that a partial degree of annealing is already sufficient to stabilize a biaxial nematic structure.Quenched and annealed degrees of freedom of statistical systems are known to produce phase diagrams with a number of distinct features [1]. The ferromagnetic site-diluted Ising model provides an example of a continuous transition, in the quenched case, which turns into a first-order boundary beyond a certain tricritical point, if we consider thermalized site dilution [2]. Disordered degrees of freedom in solid compounds, as random magnets and spin-glasses, are examples of quenched disorder, which lead to well-known problems related to averages of sets of disordered free energies. In liquid systems, however, relaxation times are shorter, and the simpler problems of annealed disorder are more relevant from the physical perspective. In this paper, we show that distinctions between quenched and annealed degrees 1