  The leitmotif of this paper is the question of whether four- and higher
even-dimensional Conformal Gravities do have a Chern-Simons pedigree. We show
that Weyl gravity can be obtained as dimensional reduction of a
five-dimensional Chern-Simons action for a suitable (gauged-fixed,
tractor-like) five-dimensional AdS connection. The gauge-fixing and dimensional
reduction program admits a readily generalization to higher dimensions for the
case of certain conformal gravities obtained by contractions of the Weyl
tensor.
