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The squashed seven-sphere operator spectrum is completed by deriving thespectrum of the spin-3/2 operator. The implications of the results for the$AdS_4$ $\mathcal{N} = 1$ supermultiplets obtained from compactification ofeleven-dimensional supergravity are analysed. The weak $G_2$ holonomy plays animportant role when solving the eigenvalue equations on the squashed sphere.Here, a novel and more universal algebraic approach to the whole eigenvalueproblem on coset manifolds is provided. Having obtained full control of all theoperator spectra, we can finally determine the irreps $D(E_0, s)$ for allsupermultiplets in the left-squashed vacuum. This includes an analysis ofpossible boundary conditions. By performing an orientation flip on theseven-sphere, we also obtain the full spectrum for the non-supersymmetricright-squashed compactification which is of interest in the swampland contextand in particular for the $AdS$ swampland conjecture. Here, a number ofboundary condition ambiguities arise making the analysis of dual marginaloperators somewhat involved. This work is a direct continuation of [1] and [2].

The complete Kaluza-Klein spectra of $\mathcal{N} = 1$ and $\mathcal{N} = 0$ M-theory on $AdS_4 \times (\text{squashed } S^7)$