  The landscape of possible four-dimensional low-energy effective theories
arising from compactifications of string/M-theory seems vast. This might lead
one to believe that any consistent-looking effective field theory coupled to
gravity can be obtained as a low-energy limit of string theory. However, a set
of "swampland" conjectures suggests that this is not true and that, in fact,
there is an even larger set of effective field theories that cannot be obtained
in this way. In particular, the AdS instability swampland conjecture asserts
that nonsupersymmetric anti-de Sitter vacua are unstable. These swampland
criteria can have implications for, for instance, low-energy physics and
cosmology.
  M-theory is a nonperturbative unification of all superstring theories. Its
low-energy limit, eleven-dimensional supergravity, admits two compactifications
on the squashed seven-sphere. One of the solutions has one unbroken
supersymmetry ($\mathcal{N} = 1$) while the other has none ($\mathcal{N} = 0$).
Due to the AdS instability swampland conjecture, the latter should be unstable.
However, this has not been demonstrated explicitly. To study the stability of
the $\mathcal{N} = 0$ vacuum, we investigate the mass spectrum of the theory.
The main advancement compared to previous attempts is the realisation that all
mass operators in the Freund-Rubin compactification are related to a universal
Laplacian, allowing us to relate Weyl tensor terms to group invariants. One
limitation of the group-theoretical method we employ is that it can lead to
false roots. This is remedied, at least in part, by demanding that the fields
form supermultiplets in the $\mathcal{N} = 1$ case. Although we arrive at an
eigenvalue spectrum for all operators of interest, there is a hint that the
results may be incomplete. Thus, we do not reach a decisive conclusion
regarding the investigated type of instability.
