The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{****}$, $\dots$ are Grothendieck spaces.
(See, e.g., this paper for the proof of Grothendieck property of these even duals. Their proof uses the fact that these spaces are C(K) spaces for large Stonian K-s.)
My question is: Is it known, in general, if the second dual of a Grothendieck space is a Grothendieck space too? Is there a counterexample?