  The "$\phi^2$" slow roll inflation combined with General Relativity is
largely excluded by Planck data. In this paper, we consider the same potential
combined with the $R+C^2$ gravity of purely virtual particles (or fakeons),
where the would-be ghost introduced by the Weyl tensor term, $C^2$, is
quantized with the fakeon prescription. We compute the tensor power spectrum in
the full theory by means of the Cosmic Renormalization Group formalism and
critically examine its physical meaning. In particular, we show that it is not
possible to retrieve the power spectrum of the fakeon free-theory by
considering the decoupling limit of the purely virtual particles. We provide a
physical explanation in terms of the causal structure of the theory to infer
that a model of quadratic inflation from purely virtual particles is also
discarded from a phenomenological point of view.
