  We explore the embedding of the phenomenological holographic models
describing thermal relativistic ordered conformal phases in ${\mathbb R}^{2,1}$
in SUGRA/String theory. The dual black branes in a Poincare patch of
asymptotically $AdS_4$ have ``hair'' -- a condensate of the order parameter for
the broken symmetry. In a gravitational dual the order parameter for a
spontaneous symmetry breaking is represented by a bulk scalar field with a
nontrivial potential. To construct the ordered conformal phases perturbatively
we introduce a phenomenological deformation parameter in the scalar potential.
We find that while the ordered phases exist for different values of the
deformation parameter, they disappear before the deformation is removed, in one
case once the potential is precisely as in the top-down holography. It appears
that the holographic models with the conformal ordered phases are in the String
theory swampland.
