Metricity conjecture for the geometric mean distance on a slit Euclidean space

Let $n$ be the ambient dimension, let $u,v\in\mathbb{R}^n$ be distinct points, and let $[u,v]$ denote the line segment joining them. For a parameter $c$, write $h_{\mathbb{R}^n\setminus[u,v],c}$ for the geometric mean distance function on the domain $\mathbb{R}^n\setminus[u,v]$. Metricity conjecture. The function $h_{\mathbb{R}^n\setminus[u,v],c}$ is a metric if and only if $c\geq1$. Numerical tests suggest this result, while the preceding discussion establishes the corresponding criterion in the hyperbolic space; the asserted metricity characterization for the complement of a line segment remains unproved in the supplied text.