  We use computer simulations to study the existence and stability of a biaxial
nematic $N_b$ phase in systems of hard polyhedral cuboids, triangular prisms,
and rhombic platelets, characterized by a long ($L$), medium ($M$), and short
($S$) particle axis. For all three shape families, we find stable $N_b$ states
provided the shape is not only close to the so-called dual shape with $M =
\sqrt{LS}$ but also sufficiently anisotropic with $L/S>9,11,14, 23$ for rhombi,
prisms, and cuboids, respectively, corresponding to anisotropies not considered
before. Surprisingly, a direct isotropic-$N_b$ transition does not occur in
these systems due to a destabilization of $N_b$ by a smectic (for cuboids and
prisms) or a columnar (for platelets) phase at small $L/S$, or by an
intervening uniaxial nematic phase at large $L/S$. Our results are confirmed by
a density functional theory provided the third virial coefficient is included
and a continuous rather than a discrete (Zwanzig) set of particle orientations
is taken into account.
