  We derive the equilibrium phase diagram of the classical dipolar Ising
antiferromagnet at the mean-field level on a geometry that mimics the two
dimensional Kagome lattice. Our mean-field treatment is based on the
combination of the cluster variational Bethe-Peierls formalism and the cavity
method, developed in the context of the glass transition, and is complementary
to the Monte Carlo simulations realized in [Phys. Rev. B 98, 144439 (2018)].
Our results confirm the nature of the low temperature crystalline phase which
is reached through a weakly first-order phase transition. Moreover, they allow
us to interpret the dynamical slowing down observed in the work of Hamp & al.
as a remnant of a spin glass transition taking place at the mean-field level
(and expected to be avoided in 2 dimensions).
