Say that a Banach space $X$ is strongly saturated by reflexive subspaces if every closed subspace $Y\subset X$ contains a further reflexive subspace $Z\subset Y$ with $\mbox{dens }Y=\mbox{dens Z}$. If I recall correctly, the long James space has this property (it is strongly saturated by Hilbert spaces).
I would like to know how far from a reflexive space can be a space with this property. To quantify "the distance to a reflexive space" I ask the following question:
Is there a Banach space $X$ strongly saturated by reflexive subspaces such that $\ell_\infty(\omega_1)$ (or $\ell_\infty / c_0$) embeds into $X^{**}$?
(in my vague understanding, containment of $\ell_\infty$, say, is still not very far...)
$X^*$' is isomorphic to $\ell_1$ (and hence$X^{**}$` is isomorphic to $\ell_\infty$). $\endgroup$