  We investigate the constraints on the superfluid fraction of an amorphous
solid following from an upper bound derived by Leggett. In order to accomplish
this, we use as input density profiles generated for amorphous solids in a
variety of different manners including by investigating Gaussian fluctuations
around classical results. These rough estimates suggest that, at least at the
level of the upper bound, there is not much difference in terms of
superfluidity between a glass and a crystal characterized by the same Lindemann
ratio. Moreover, we perform Path Integral Monte Carlo simulations of
distinguishable Helium 4 rapidly quenched from the liquid phase to very lower
temperature, at the density of the freezing transition. We find that the system
crystallizes very quickly, without any sign of intermediate glassiness. Overall
our results suggest that the experimental observations of large superfluid
fractions in Helium 4 after a rapid quench correspond to samples evolving far
from equilibrium, instead of being in a stable glass phase. Other scenarios and
comparisons to other results on the super-glass phase are also discussed.
