  We compute anomalous dimensions of quartic operators which are singlets under
the $\mathrm{U}(N_f)$ global symmetry in Yang-Mills theories with Chern-Simons
level $k$ in three dimensions coupled to $N_f$ Dirac fermions. In order to have
analytic control, we consider the regime $N_f\gg N_c\gg 1$, where the problem
is reduced to the study of a flavor-adjoint and a flavor-singlet bilinears
whose square give the quartic operators of interest. We provide evidence that
these operators hit marginality, signaling instabilities which, for
$\frac{2k}{N_f}<1$ suggest the spontaneous breaking of the global symmetry, and
no symmetry breaking otherwise. For $k=N_f/2-1$ (the value corresponding to the
domain walls of 4d QCD at $\theta=\pi$), the critical value $N_f^*$ is
tantalizingly close to the lower end of the conformal window of QCD$_4$,
suggesting a connection between conformal and global symmetry breaking in the
4d theory and in its domain walls. We also study, at $k=0$, other quartic
operators containing a singlet when branched under
$\mathrm{U}\left(\frac{N_f}{2}\right)\times
\mathrm{U}\left(\frac{N_f}{2}\right)$, finding that they hit marginality
precisely at the same point as their flavor-neutral cousins. Using the same
technology we study bosonic CS-QCD$_3$, finding no hint of symmetry breaking
where our analysis is applicable.
