Ni's maximal volume growth conjecture for Kähler manifolds

Let $M^n$ be a complete noncompact Kähler manifold with nonnegative bisectional curvature. Suppose that $M$ admits a nonconstant holomorphic function with polynomial growth and that its bisectional curvature is positive at least at one point. Ni's maximal volume growth conjecture. Then $M$ has maximal volume growth. More precisely, if $M$ has quasi-positive bisectional curvature, then the following three conditions are equivalent: $\mathcal{O}_P(M)\neq\mathbb{C}$, average quadratic curvature decay, and maximal volume growth. Average quadratic curvature decay means that, for every $r>0$,

$$\Xint-_{B(p,r)} S\leq \frac{C}{r^2},$$

where $p$ is a point on $M$, $C$ is a positive constant independent of $r$, and $S$ is the scalar curvature. Here $\Xint-$ denotes the average. This conjecture concerns the relationship between holomorphic functions of polynomial growth, curvature decay, and volume growth in complete Kähler manifolds with nonnegative bisectional curvature; the supplied source does not establish its resolution.