  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 $\ell_1$,
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.
