On decompositions of Banach spaces of continuous functions on Mrówka's spaces

It is well known that if $K$ is infinite compact Hausdorff and scattered (i.e., with no perfect subsets), then the Banach space $C(K)$ of continuous functions on $K$ has complemented copies of $c_{0}$, i.e., $C(K)\sim c_{0} \oplus X\sim c_{0}\oplus c_{0} \oplus X\sim c_{0}\oplus C(K)$. We address the question if this could be the only type of decompositions of $C(K)\not \sim c_{0}$into infinite-dimensional summands for $K$ infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martin's axiom we construct an example of Mrówka's space (i.e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.