Asymptotically symmetric spaces with hereditarily non-unique spreading models

We examine a variant of a Banach space X0,11\mathfrak {X}^1_{0,1} defined by Argyros, Beanland, and the second-named author that has the property that it admits precisely two spreading models in every infinite dimensional subspace. We prove that this space is asymptotically symmetric and thus it provides a negative answer to a problem of Junge, the first-named author, and Odell.