We find the conditions under which scale-invariant Einstein-Cartan gravity with scalar matter fields leads to an approximate conformal invariance of the flat space particle theory up to energies of the order of the Planck mass. In the minimal setup, these models, in addition to the fields of the Standard Model and the graviton, contain only one extra particle - a massless dilaton. Theories of this type can pave the way for a self-completion all the way up the Planck scale and lead to rather universal inflationary predictions, close to those of the simplest Higgs-inflation scenario in the metric theory of gravity.