The point pair function quasi-metric constant conjecture

Let $G e\rrbracket\mathbb{R}^n$ be a domain, and let $p_G$ denote the point pair function. A function $d$ is a quasi-metric if there is a constant $c\geq 1$ such that $d(x,y)\leq c(d(x,z)+d(z,y))$ for all points in its domain. The point pair function quasi-metric constant conjecture. For all domains $G\subsetneq\mathbb{R}^n$, the point pair function $p_G$ is a quasi-metric with a constant less than or equal to $\sqrt{5}/2$. The preceding result gives the general bound $\sqrt{2}$, while numerical tests suggest that the sharper universal constant $\sqrt{5}/2$ is valid; the exact optimal constant remains open.