  In a recent paper Hilhorst \cite{Hilhorst2010} illustrated that the
$q$-Fourier transform for $q>1$ is not invertible in the space of density
functions. Using an invariance principle he constructed a family of densities
with the same $q$-Fourier transform and claimed that $q$-Gaussians are not
mathematically proved to be attractors. We show here that none of the
distributions constructed in Hilhorst's counterexamples can be a limit
distribution in the $q$-CLT, except the one whose support covers the whole real
axis, which is precisely the $q$-Gaussian distribution.
