A Banach space $X$ has property (V) whenever for each Banach space $Y$, every unconditionally converging operator $T:X\to Y$ is weakly compact; equivalently, every non-weakly compact operator $T:X\to Y$ is an isomorphism on a subspace of $X$ isomorphic to $c_0$.
The space $X$ has the Grothendieck property whenever for each separable Banach space $Y$, every operator $T:X\to Y$ is weakly compact.
Exercise VII.12 in J. Diestel's book "Sequences and series in Banach spaces" (Springer 1984) asks to prove that, for a dual space $X^*$, property (V) implies the Grothendieck property. It suggest to keep in mind Phillips's lemma.
Can anyone suggest an argument or a reference for the proof?
Note: Since separable spaces with the Grothendieck property are reflexive, the space $c_0$ shows that the result fails for non-dual spaces.