On the complete separation of unique ${\mathrm{\ell }}_1$ spreading models and the Lebesgue property of Banach spaces

We construct a reflexive Banach space $X_{\mathcal{D}}$ with an unconditional basis such that all spreading models admitted by normalized block sequences in $X_{\mathcal{D}}$ are uniformly equivalent to the unit vector basis of ${\mathrm{\ell }}_1$, yet every infinite-dimensional closed subspace of $X_{\mathcal{D}}$ fails the Lebesgue property. This is a new result in a program initiated by Odell in 2002 concerning the strong separation of asymptotic properties in Banach spaces.