  A recent result of Freeman, Odell, Sari, and Zheng states that whenever a
separable Banach space not containing $\ell_1$ has the property that all
asymptotic models generated by weakly null sequences are equivalent to the unit
vector basis of $c_0$ then the space is Asymptotic $c_0$. We show that if we
replace $c_0$ with $\ell_1$ then this result is no longer true. Moreover, a
stronger result of B. Maurey - H. P. Rosenthal type is presented, namely, there
exists a reflexive Banach space with an unconditional basis admitting $\ell_1$
as a unique asymptotic model whereas any subsequence of the basis generates a
non-Asymptotic $\ell_1$ subspace.
