  In this paper, we continue our investigations of R\'acz's
parabolic-hyperbolic formulation of the Einstein vacuum constraints. Our
previous studies of the asymptotically flat setting provided strong evidence
for unstable asymptotics which we were able to resolve by introducing a certain
modification of R\'acz's parabolic-hyperbolic formulation. The primary focus of
the present paper here is the asymptotically hyperboloidal setting. We provide
evidence through a mixture of numerical and analytical methods that the
asymptotics of the solutions of R\'acz's parabolic-hyperbolic formulation is
stable, and, in particular, no modifications are necessary to obtain solutions
that are asymptotically hyperboloidal.
