It was proved by W.B. Johnson and H.P. Rosenthal [[Studia Math. 43 (1972), 77–92]](http://matwbn.icm.edu.pl/ksiazki/sm/sm43/sm4317.pdf)
that every Banach space $X$ with $X^{\*\*}$ separable is *hereditarily reflexive*:
every infinite dimensional closed subspace of $X$ contains an infinite dimensional
reflexive subspace.


Suppose that $X$ is separable and $X^{\*\*}/X$ reflexive. Is $X$ hereditarily reflexive?


Of course, we would have a positive answer if each infinite dimensional closed subspace of such a space $X$ contains an infinite dimensional subspace $Y$ with $Y^{\*\*}$ separable.



On hereditarily reflexive Banach spaces
The question has a negative answer:


Following the idea in Bill Johnson's comment, I looked at the work of Argyros. In this [**paper**](https://arxiv.org/abs/0807.2392) (see the reference below), there are several examples of hereditarily indecomposable Banach spaces containing no infinite dimensional reflexive subspaces.


One of these spaces $X$ has the additional property that $X^{\*\*}/X$ is isomorphic to $\ell\_2(\Gamma)$ with $\Gamma$ uncountable: see Proposition 5.4 in the paper.


**[reference]** S.A. Argyros, A.D. Arvanitakis, A.G. Tolias.
*Saturated extensions, the attractors method and Hereditarily James Tree Spaces.*
pp. 1-90 in "Methods in Banach space theory". J.M.F. Castillo and W.B. Johnson eds.
London Math. Soc. Lecture Note Ser. 337, Cambridge University Press, 2006.


