Metricity of the point pair function off the z-axis

Let $Z$ be the $z$-axis in $\mathbb{R}^3$, and let $G=\mathbb{R}^3\backslash Z$. The point pair function $p_G$ is defined on pairs of points in $G$. Metricity conjecture. The point pair function $p_G$ is a metric on $G$. No resolution is given in the source.