Analytic structure of the $n=7$ scattering amplitude in $\mathcal{N}=4$ SYM theory in multi-Regge kinematics: Conformal Regge cut contribution

In this second part of our investigation of the analytic structure of the $2\to 5$ scattering amplitude in the planar limit of $\mathcal{N}=4$ SYM in multi-Regge kinematics we compute, in all kinematic regions, the Regge cut contributions in leading order. The results are infrared finite and conformally invariant.