Geometric conditions which imply $w^*$-normal structure

Let $X$ be a complex Banach space. In their seminal work, Lau and Mah (\cite{LM}) study a geometric property called Lim's condition which, for separable $X$, implies that for any non-empty $w^*$-compact convex set $K \subset X^*$, any nonexpansive mapping $T: K \rightarrow K$ has a fixed point. In