  In this paper we consider the hyperbolic formulation of the constraints
introduced by R\'acz. Using the numerical framework recently developed by us we
construct initial data sets which can be interpreted as nonlinear perturbations
of Schwarzschild data in Kerr-Schild coordinates and investigate their
asymptotics. Our results suggest that, unless one finds a way to exploit the
freedom to pick the free part of the initial data in some suitable way, generic
initial data sets obtained by this method may violate fundamental asymptotic
conditions.
