Nonmaximal symmetry breaking patterns in the supersymmetric $\widehat {\mathfrak{s} \mathfrak{u}}(8)_{k_U =1}$ theory

We study the nonmaximal symmetry breaking patterns of ${\mathfrak{s} \mathfrak{u}}(8) \to \mathfrak{g}_{531}/\mathfrak{g}_{351}$ allowed in a supersymmetric theory with the affine Lie algebra of $\widehat {\mathfrak{s} \mathfrak{u}}(8)_{k_U=1}$, which was recently proposed to explain the three-generational standard model quark/lepton mass hierarchies as well as the Cabbibo-Kobayashi-Maskawa mixing pattern. Three gauge couplings are found to achieve the unification below the Planck scales, according to the unification relations determined by the affine Lie algebra at level of $1$. We also prove that a separate nonmaximal symmetry breaking pattern of ${\mathfrak{s} \mathfrak{u}}(8) \to \mathfrak{g}_{621}$ is unrealistic due to the additional massless vectorlike quarks in the spectrum.