  The phase diagram of a polydisperse mixture of uniaxial rod-like and
plate-like hard parallelepipeds is determined for aspect ratios $\kappa=5$ and
15. All particles have equal volume and polydispersity is introduced in a
highly symmetric way. The corresponding binary mixture is known to have a
biaxial phase for $\kappa=15$, but to be unstable against demixing into two
uniaxial nematics for $\kappa=5$. We find that the phase diagram for
$\kappa=15$ is qualitatively similar to that of the binary mixture, regardless
the amount of polydispersity, while for $\kappa=5$ a sufficient amount of
polydispersity stabilizes the biaxial phase. This provides some clues for the
design of an experiment in which this long searched biaxial phase could be
observed.
