  We construct a Weyl-gauging of strongly coupled Einsteinian-Cubic gravity
(ECG) in Weyl's geometry via upgraded Weyl's curvature tensors, which include
abelian gauge fields, and properly tuned compensating real scalar fields. The
model is free from any dimensionful parameters. The bare ECG emerges as the
lower energy limit of the Weyl-ECG in the local non-conformal-invariant vacua
(i.e., broken phase) in the maximally symmetric spacetimes fixing the vacuum
expectation value of the scalar field to the Planck mass scale. Here, the
natural presence of (anti-) de Sitter backgrounds spontaneously breaks Weyl's
local conformal symmetry akin to the Higgs mechanism, while it is radiatively
broken at the renormalization scale at the one-loop level in flat vacua through
the Coleman-Weinberg mechanism. The model allows anti-de Sitter and flat spaces
but does not allow de Sitter to be vacuum spacetime solutions. The properties
of the model deserve further exploration, specifically, those of
nonperturbative (e.g., instantons and/or anti-instantons) contributions, for
example, in the resurgence or tachyon condensation context requires detailed
study.
