Weak*-complemented subspaces of $\ell_\infty$

Question

Consider $\ell_\infty$ as $\ell_1^*$. Let $X$ be an infinite-dimensional complemented subspace of $\ell_\infty$ (in partiuclar, $X$ is isomorphic to $\ell_\infty$). Can we find a further subspace $Y\subset X$ which is still isomorphic to $\ell_\infty$ but it is also range of a weak*-weak* continuous projection on $\ell_\infty$?