It is well-known that non-weakly compact operators from $\ell_\infty$ into any Banach space act as isomorphisms on some subspace of $\ell_\infty$ isomorphic to $\ell_\infty$. I have a question in this spirit.

Suppose that $T\colon \ell_\infty \to c_0$ is a bounded linear operator. Then $T$ is not weakly compact. Can it happen that $T|_{c_0}\colon c_0\to c_0$ is weakly compact?