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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 thatwhenever a Banach space not containing $\ell_1$, in particular a reflexiveBanach 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 relationbetween specific asymptotic structures in Banach spaces, initiated in aprevious paper by the first and fourth authors. For the definition of$\mathfrak{X}^{(p)}_{\text{awi}}$ we use saturation with asymptotically weaklyincomparable constraints, a new method for defining a norm that remains smallon a well-founded tree of vectors which penetrates any infinite dimensionalclosed subspace.

The complete separation of the two finer asymptotic $\ell_{p}$ structures for $1\le p<\infty$