Completeness conjecture for Kähler–Einstein metrics near isolated log canonical singularities

Let $(\mathcal{U},p)$ be the germ of an isolated log canonical, non-log-terminal singularity, and let $g_{KE}$ be a Kähler–Einstein metric constructed in Theorem 1.1 on $\mathcal{U}\setminus\{p\}$. For a fixed point $q\in\mathcal{U}\setminus\{p\}$ and $R>0$, consider paths in $\mathcal{U}\setminus\{p\}$ joining $q$ to points approaching $p$.

Completeness conjecture. The metric $g_{KE}$ is complete near $p$; more precisely, there is an open neighborhood $V_{q,R}$ of $p$ such that for every $x\in V_{q,R}\setminus\{p\}$ and every smooth path $\gamma$ joining $q$ and $x$ in $\mathcal{U}\setminus\{p\}$,

$$|\gamma|_{g_{KE}}>R,$$

where $|\gamma|_{g_{KE}}$ denotes the arc length of $\gamma$ with respect to $g_{KE}$.

This conjecture asks whether the finite-volume Kähler–Einstein metric constructed near an isolated log canonical singularity is nevertheless metrically complete toward the singular point. The paper proposes this as the geometric question governing the behavior of the metric near $p$, and does not resolve it in general.