A Banach space $E$ is called Grothendieck if every weak* convergent sequence in the dual space $E^*$ is weakly convergent. A typical example of a Grothendieck space is $\ell_\infty$. Diestel proved the following characterization of Grothendieck Banach spaces:
For every Banach space $E$, TFAE:
- $E$ is Grothendieck,
- for each separable Banach space $F$, every bounded operator $T\colon E\to F$ is weakly compact,
- every bounded operator $T\colon E\to c_0$ is weakly compact.
My question: Can we replace the space $c_0$ in item 3 by some/any other fixed separable space? In particular, can we use here $\ell_1$?
(By "other" I of course mean "non-isomorphic to $c_0$".)