  The existence of superfluidity of the neutron component in the core of a
neutron star, associated specifically with triplet $P-$wave pairing, is
currently an open question that is central to interpretation of the observed
cooling curves and other neutron-star observables. Ab initio theoretical
calculations aimed at resolving this issue face unique challenges in the
relevant high-density domain, which reaches beyond the saturation density of
symmetrical nuclear matter. These issues include uncertainties in the
three-nucleon (3N) interaction and in the effects of strong short-range
correlations -- and more generally of in-medium modification of nucleonic
self-energies and interactions. A survey of existing solutions to the gap
equations in the triplet channel shows that the separate or combined impacts of
3N forces, coupled channels, and mass renormalization range from moderate to
strong to devastating, thus motivating a detailed analysis of the competing
effects. In the present work we track the effects of the 3N force and in-medium
modifications in the representative case of the $^3P_2$ channel, based on the
Argonne V18 two-nucleon (2N) interaction supplemented by 3N interactions of the
Urbana IX family. Sensitivity of the results to the input interaction is
clearly demonstrated, while consistency issues arise with respect to the
simultaneous treatment of 3N forces and in-medium effects. We consider this
pilot study as the first step towards a systematic and comprehensive
exploration of coupled-channel $^3P F_2$ pairing using a broad range of 2N and
3N interactions from the current generation of refined semi-phenomenological
models and models derived from chiral effective field theory.
