  Recent work on flux compactifications suggests that the tadpole constraint
generically allows only a limited number of complex structure moduli to become
massive, i.e., be stabilized at quadratic order in the spacetime
superpotential. We study the effects of higher-order terms systematically
around the Fermat point in the $1^9$ Landau-Ginzburg model. This model lives at
strong coupling and features no K\"ahler moduli. We show that, depending on the
flux, several massless fields can indeed be stabilized in this fashion, and
argue that this paves the way to explicit ${\mathcal N}=1$ Minkowski vacua
without flat directions. Along the way, we complete the classification of
integral flux vectors with small tadpole contribution. Thereby we are closing
in on a future complete understanding of all possible flux configurations in
the $1^9$ Landau-Ginzburg model.
