PFA and complemented subspaces of ℓ∞/c0

The Banach space $\ell_\infty/c_0$ is isomorphic to the linear space of continuous functions on $\mathbb N^*$ with the supremum norm, $C(\mathbb N^*)$. Similarly, the canonical representation of the  $\ell_\infty$ sum of $\ell_\infty/c_0$ is the Banach space of  continuous functions on the closure of any non-compact cozero subset of $\mathbb N^*$.  It is important to determine if there is a continuous linear lifting of this Banach space to a complemented subset of $C(\mathbb N^*)$. We show that PFA implies there is no such lifting.