It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space.

I would like to know if the weakened module version of this question is answered. More precisely: For which compact Hausdorff spaces $X$ the module $C(X)$ is an isomorphically injective $C(X)$-module. I know that for $X$ Stonean $C(X)$ is even an isometrically injective $C(X)$-module.