On hereditarily reflexive Banach spaces

Question

It was proved by W.B. Johnson and H.P. Rosenthal [Studia Math. 43 (1972), 77–92] 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.

Answer 1

The question has a negative answer:

Following the idea in Bill Johnson's comment, I looked at the work of Argyros. In this paper (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.