Odaka's asymptotics conjecture for complete Ricci-flat Kähler metrics

Let $X^{o}$ be a non-proper separated complex variety with at most Kawamata log terminal singularities, carrying a complete Ricci-flat weak Kähler metric $g$ and a base point $p$. Suppose

$$\operatorname{vol}(B(p,r))\sim c r^{d}$$

as $r\to+\infty$, and define the volume growth dimension by ${\rm vg}(X^{o},g)=d$. If ${\rm vg}(X^{o},g)>\dim_{\mathbb C}(X)$, let $\bar\kappa(X^{o})$ denote the logarithmic Iitaka dimension; if $X^{o}=X\setminus\operatorname{Supp}(D)$ for a smooth projective $X$ and a simple normal crossing divisor $D\in|-K_X|$, let $\widetilde\Delta(D)$ denote the dual intersection cone complex and $(X^{o})^{\rm an}$ its Berkovich analytification for the trivial valuation. Asymptotics conjecture. (1) If ${\rm vg}(X^{o},g)>\dim_{\mathbb C}(X)$, then $\bar\kappa(X^{o})=-\infty$, and every compactification $X$ is uniruled. (2) In the stated log-smooth anticanonical compactification case,

$$\dim_{\mathbb R}\widetilde\Delta(D)=\dim (X^{o})^{\rm an}\le {\rm vg}(X^{o},g)\le \dim_{\mathbb R}(X)=2\dim_{\mathbb C}(X).$$

These expected relations connect metric volume growth with logarithmic birational geometry and the dual complex; the source gives no resolution.