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.