User blog comment:Upquark11111/An Explanation of Loader's Number/@comment-11227630-20171209072636/@comment-11227630-20171210001134

However, the extra condition $$\Gamma\vdash A:s_1$$ is so important that the choice of $$s_1$$ is where the 8 systems differ. (For example, if we only allow $$s_1=*$$ the system will become λP instead of λPω.) Without this condition, Loader's system might be too strong (stronger than all the 8 λ-cube systems) - it might not have normalization property, and it might be inconsistent, which make Loader's program fail.