  We study quantum multicritical behavior in a (2+1)-dimensional
Gross-Neveu-Yukawa field theory with eight-component Dirac fermions coupled to
two triplets of order parameters that act as Dirac masses, and transform as
$(1,0) + (0,1)$ representation under the SO(4)$\simeq$SO(3)$\times$SO(3)
symmetry group. This field theory is relevant to spin-1/2 fermions on honeycomb
or $\pi$-flux lattices, for example, near the transition points between an
$s$-wave superconductor and a charge-density wave, on one side, and N\'eel
order, on the other. Two triplets of such order parameters always allow for a
common pair of two other order parameters that would complete them to the
maximal set of compatible (anticommuting) orders of five. We first derive a
unitary transformation in the Nambu (particle-hole) space which maps any two
such triplets, possibly containing some superconducting orders, onto purely
insulating order parameters. This allows one to consider a universal SO(4)
Gross-Neveu-Yukawa description of the multicriticality without any Nambu
doubling. We then proceed to derive the renormalization-group flow of the
coupling constants at one-loop order in $4-\epsilon$ space-time dimensions,
allowing also a more general set of order parameters transforming under
SO($n_a$)$\times$SO($n_b$). While for $n_a=n_b > 2 $ in the bosonic sector and
with fermions decoupled there is a stable fixed point of the flow, the Yukawa
coupling to fermions quickly leads to its elimination by a generic fixed-point
collision in the relevant range of fermion flavor numbers $N_f$. This suggests
the replacement of the critical behavior by a runaway flow in the physical case
$n_a=n_b=3$. The structure of the RG flow at $n_a\neq n_b$ is also discussed,
and some non-perturbative arguments in favor of the stability of the decoupled
critical point when $n_a=3$ and $n_b=1$ in $D=2+1$ are provided.
