Let $(M, g)$ be a (complete) Kähler manifold with Ricci curvature $\geq c$.

Is it true that the volume ratio of geodesic balls in $M$ with respect to balls in the corresponding (simply connected) complex space form with Ricci $\equiv c$ is a decreasing function?

I suspect the answer is no. Examples would be nice, also showing why the Riemannian proof breaks would help.

EDIT: A friend made me notice Gang Liu's paper (https://arxiv.org/pdf/1108.4231v1.pdf), where it is shown that, for real analytic metrics, the volume ratio is decreasing for small values of the radius.