  We modify the conformal inflation set-up of \cite{Kallosh:2013hoa}, with a
local Weyl invariance, a dynamical Planck scale and an $SO(1,1)$ invariance at
high energies, in order to be able to identify the physical scalar with the
physical Higgs at low energies, similar to the cyclic Higgs models of
\cite{Bars:2013vba}. We obtain a general class of exponentially-corrected
potentials that gives a generalized Starobinsky model with an infinite series
of $R^p$ corrections, tracing a line in the $(r,n_s)$ plane for CMB
fluctuations, with an $r$ that can be made to agree with the recent BICEP2
measurement. Introducing a coupling different from the conformal value, thus
breaking the local Weyl symmetry, leads nevertheless to a very strong attractor
motion towards the generalized Starobinsky line.
