  In this paper, we proceed as suggested in the final section of
arXiv:1812.03874v2 and prove a lower bound for the spectral gap of the
conjugate Kac process with 3 interacting particles. This bound turns out to be
around $0.02$, which is already physically meaningful, and we perform Monte
Carlo simulations to provide a better empirical estimate for this value via
entropy production inequalities. This finishes a complete quantitative estimate
of the spectral gap of the Kac process.
