Exact $\mathscr K$-monotonicity of one class of Banach pairs

We obtain necessary and sufficient conditions for exact $\mathscr K$-monotonicity of Banach pairs which are constituted by the space of essentially bounded functions and an arbitrary Lorentz space. The proof bases on a description for the set of extreme points of $\mathscr K$-orbits with respect to the corresponding finite-dimensional pairs.