Subprojectivity of projective tensor products of Banach spaces of continuous functions

Galego and Samuel showed that if $K,L$ are metrizable, compact, Hausdorff spaces, then $C(K){\widehat{\otimes }}_{\pi }C(L)$ is $c_0$-saturated if and only if it is subprojective if and only if $K$ and $L$ are both scattered. We remove the hypothesis of metrizability from their result, and extend it from the case of the two-fold projective tensor product to the general $n$-fold projective tensor product to show that for any $n\in \mathbb{N}$ and compact, Hausdorff spaces $K_1,\dots ,K_n$, ${\widehat{\otimes }}_{\pi ,i=1}^{n}C(K_i)$ is $c_0$-saturated if and only if it is subprojective if and only if each $K_i$ is scattered.