  Biaxial nematic ($N_B$) liquid crystals have been indicated as promising
candidates for the design of next-generation displays with novel
electro-optical properties and faster switching times. While at the molecular
scale their existence is still under debate, experimental evidence, supported
by theory and simulation, has unambiguously proved that suitable colloidal
particles can indeed form $N_B$ fluids under specific conditions. While this
discovery has sparked a widespread interest in the characterisation of the
phase behaviour of $N_B$ liquid crystals, significantly less attention has been
devoted to the study of their transport properties. To bridge this gap, by
Dynamic Monte Carlo simulations we have investigated the equilibrium dynamics
of field-induced $N_B$ phases comprising monodisperse hard cuboids. In
particular, we calculated the long-time self-diffusion coefficients of cuboids
over a wide range of anisotropies, spanning prolate to oblate geometries.
Additionally, we have compared these diffusivities with those that, upon
switching the external field off, are measured in the thermodynamically-stable
isotropic or uniaxial nematic phases at the same density. Our results indicate
that while prolate cuboids diffuse significantly faster in biaxial nematics
than in less ordered fluids, we do not observe such an increase with oblate
cuboids at high packing fractions. We show that these changes are most likely
due to the field-induced freezing of the axes perpendicular to the nematic
director, along with a substantial increase in the ordering of the resulting
$N_B$ phase.
