  The gravitational entropy and no-hair conjectures seems to predict
contradictory future states of our Universe. The growth of the gravitational
entropy is associated with the growth of inhomogeneity, while the no-hair
conjecture argues that a universe dominated by dark energy should
asymptotically approach a homogeneous and isotropic de Sitter state. The aim of
this paper is to study these two conjectures. The investigation is based on the
Simsilun simulation, which simulates the universe using the approximation of
the Silent Universe. The Silent Universe is a solution to the Einstein
equations that assumes irrotational, non-viscous, and insulated dust, with
vanishing magnetic part of the Weyl curvature. The initial conditions for the
Simsilun simulation are sourced from the Millennium simulation, which results
with a realistically appearing but relativistic at origin simulation of a
universe. The Simsilun simulation is evolved from the early universe (t = 25
Myr) till far future (t = 1000 Gyr). The results of this investigation show
that both conjectures are correct. On global scales, a universe with a positive
cosmological constant and non-positive spatial curvature does indeed approach
the de Sitter state. At the same time it keeps generating the gravitational
entropy.
