  The stability of demixing phase transition in binary mixtures of hard plates
(with thickness L and diameter D) and hard spheres (with diameter $\sigma$) is
studied by means of Parsons-Lee theory. The isotropic-isotropic demixing, which
is found in mixtures of large spheres and small plates, is very likely to be
preempted by crystallization. In contrast, the nematic-nematic demixing, which
is obtained in mixtures of large plates and small spheres, can be stabilized at
low diameter ratios ($\sigma$/D) and aspect ratios (L/D). At intermediate
values of $\sigma$/D, where the sizes of the components are similar, neither
the isotropic-isotropic nor the nematic-nematic demixing can be stabilized, but
a very strong fractionation takes place between a plate rich nematic and a
sphere rich isotropic phases. Our results show that the excluded volume
interactions are capable alone to explain the experimental observation of the
nematic-nematic demixing, but they fail for the description of
isotropic-isotropic one (Chen et. al., Soft Matter, 11, 5775 (2015)).
