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

High Energy Physics - Theory