Yang's refined minimal-degree conjecture for Kähler manifolds

Let $(M^n,g)$ be a complete noncompact Kähler manifold with nonnegative bisectional curvature, and assume that the universal cover of $M$ does not split. For a point $p$, let $\overrightarrow{d_{\operatorname{min}}}(p)=(d_{\operatorname{min}}^{(1)},\ldots,d_{\operatorname{min}}^{(n)})$ denote the refined minimal-degree vector defined using $n$-tuples of global polynomial-growth holomorphic functions in $\mathcal{O}_P(M,p)$ that give local coordinates at $p$, with the component growth rates arranged nondecreasingly. Let $\operatorname{AVR}(M,g)$ be the asymptotic volume ratio, $D_{\operatorname{min}}$ the minimal degree, and $\operatorname{ASCD}(M,g)$ the average of scalar-curvature decay. Yang's refined minimal-degree conjecture. If $M$ has maximal volume growth, then $\overrightarrow{d_{\operatorname{min}}}(p)$ is realized by an $n$-tuple in $\mathcal{O}_P(M,p)$ and is independent of $p$, and

$$\operatorname{AVR}(M,g)=\prod_{i=1}^n\frac{1}{d_{\operatorname{min}}^{(i)}},$$ $$D_{\operatorname{min}}=\sum_{i=1}^n d_{\operatorname{min}}^{(i)}-n,$$

and

$$\operatorname{ASCD}(M,g)=4nD_{\operatorname{min}}.$$

This is presented as a quantitative refinement of the relevant uniformization and volume-growth theorem, relating holomorphic-function growth to asymptotic volume and scalar-curvature decay. The supplied text gives no resolution of the conjecture.