  In this work, for the first time in literature, we study the predictions of
non-minimally coupled Natural and Coleman-Weinberg potentials in the $n_s-r$
plane, and an extended $\Lambda$CDM model where we include non-standard
self-interactions among massive neutrinos, mediated by a heavy scalar or vector
boson. Constraints were derived using the Planck 2018 + BICEP/Keck 2018
datasets along with other data. For the inflationary potentials, we consider
two different formulations in gravity that are non-minimally coupled to the
scalar field of the inflaton: \textit{Metric and Palatini.} We only consider
the self-interaction to be present among $\tau$-neutrinos and only at moderate
strengths. This is because strong interactions among $\tau$-neutrinos, or any
strength self-interaction among electron- and muon-neutrinos, as well as any
strength flavor-universal interactions, are strongly disfavoured from particle
physics experiments.
  In terms of cosmological data, we use the latest public CMB datasets from
Planck 2018 and BICEP/Keck 2018 collaborations, along with other data from CMB
lensing, BAO, RSD, and SNe Ia luminosity distance measurements. We find that
there are some situations where predictions from the inflationary models are
ruled out at more than 2$\sigma$ by the minimal $\Lambda$CDM$+r$ model, but
they are allowed in the self-interacting neutrino scenario.
