  Marginally stable solids have peculiar physical properties that were
discovered and analyzed in the context of the jamming transition. We
theoretically investigate the existence of marginal stability in a prototypical
model for structural glass-formers, combining analytical calculations in
infinite dimensions to computer simulations in three dimensions. While
mean-field theory predicts the existence of a Gardner phase transition towards
a marginally stable glass phase at low temperatures, simulations show no hint
of diverging timescales or lengthscales, but reveal instead the presence of
sparse localized defects. Our results suggest that the Gardner transition is
deeply affected by finite dimensional fluctuations, and raise issues about the
relevance of marginal stability in structural glasses far away from jamming.
