  Dirac spin liquids represent a class of highly-entangled quantum phases in
two dimensional Mott insulators, featuring exotic properties such as critical
correlation functions and absence of well-defined low energy quasi-particles.
Existing numerical works suggest that the spin-orbital $SU(4)$ symmetric
Kugel-Khomskii model of Mott insulators on the honeycomb lattice realizes a
Dirac spin-orbital liquid, described at low energy by $(2+1)d$ quantum
electrodynamics (QED$_3$) with $N_f=8$ Dirac fermions. We generalize methods
previously developed for $SU(2)$ spin systems to analyze the symmetry
properties and stability of the Dirac spin-orbital liquid. We conclude that the
standard Dirac state in the $SU(4)$ honeycomb system, based on a simple parton
mean-field ansatz, is intrinsically unstable at low energy due to the existence
of a monopole perturbation that is allowed by physical symmetries and relevant
under renormalization group flow. We propose two plausible alternative
scenarios compatible with existing numerics. In the first scenario, we propose
an alternative $U(1)$ Dirac spin-orbital liquid, which is similar to the
standard one except for its monopole symmetry quantum numbers. This alternative
$U(1)$ state represents a stable gapless phase. In the second scenario, we
start from the standard $U(1)$ Dirac liquid and Higgs the $U(1)$ gauge symmetry
down to $\mathbb{Z}_4$. The resulting $\mathbb{Z}_4$ Dirac spin-orbital liquid
is stable. We also discuss the continuous quantum phase transitions from the
$\mathbb{Z}_4$ Dirac liquids to conventional symmetry-breaking orders,
described by the QED$_3$ theory with $N_f=8$ supplemented with a critical
charge-$4$ Higgs field. We discuss possible ways to distinguish these scenarios
in numerics. We also extend previous calculations of the quantum anomalies of
QED$_3$ and match with generalized lattice Lieb-Schultz-Mattis constraints.
