In the study of the asymptotic structure of Banach spaces, strong assumptions concerning the asymptotic geometry of a space may imply facts about the original geometry. Spreading models are one common tool in this regard. For example, assume that $X$ is a space with a basis $(x_i)$ such that every spreading model of a normalized block sequence of $(x_i)$ is 1-equivalent to some fixed basic sequence $(e_i)$. Does $X$ contain $[e_i]$ isomorphically? The answer to this question is known to be yes whenever $(e_i)$ is the unit vector basis of $\ell_1$ or $c_0$. It has been asked if the same is true of $\ell_p$ for $1<p<\infty$. We present some partial results on this question including a positive result under the additional assumption that all the normalized block sequences give rise to spreading models without passing to subsequences
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