  For $1\le p <\infty$, we present a reflexive Banach space
$\mathfrak{X}^{(p)}_{\text{awi}}$, with an unconditional basis, that admits
$\ell_p$ as a unique asymptotic model and does not contain any Asymptotic
$\ell_p$ subspaces. D. Freeman, E. Odell, B. Sari and B. Zheng have shown that
whenever a Banach space not containing $\ell_1$, in particular a reflexive
Banach space, admits $c_0$ as a unique asymptotic model then it is Asymptotic
$c_0$. These results provide a complete answer to a problem posed by L.
Halbeisen and E. Odell and also complete a line of inquiry of the relation
between specific asymptotic structures in Banach spaces, initiated in a
previous paper by the first and fourth authors. For the definition of
$\mathfrak{X}^{(p)}_{\text{awi}}$ we use saturation with asymptotically weakly
incomparable constraints, a new method for defining a norm that remains small
on a well-founded tree of vectors which penetrates any infinite dimensional
closed subspace.
