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.
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