Self-conjugate representation SU(3) chains

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 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.