  It was recently argued that SU(3) chains in the $p$-box symmetric irreducible
representation (irrep) exhibit a "Haldane gap" when $p$ is a multiple of 3 and
are otherwise gapless [Nucl. Phys. B $\boldsymbol{924}$, 508 (2017)]. We extend
this argument to the self-conjugate irreps of SU(3) with $p$ columns of length
2 and $p$ columns of length 1 in the Young tableau ($p=1$ corresponding to the
adjoint irrep), arguing that they are always gapped but have spontaneously
broken parity symmetry for $p$ odd but not even.
