An asymptotic property of Schachermayers space under renorming

A Banach space X with closed unit ball B is said to have property 2-beta, repsectively 2-NUC if for every ep > 0, there exists delta > 0 such that for every ep-separated sequence (x_n) in the unit ball B, and every x in B, there are distinct indices m and n such that ||x + x_m + x_n|| < 3(1 - delta), respectively, ||x_m + x_n|| < 2(1 - delta). It is shown that a Banach space constructed by Schachermayer has property 2-beta but cannot be renormed to have property 2-NUC.