AMRZ conjecture on graph lines and universal pairs

Let $G=(V,E)$ be a connected graph with at least two vertices. Let $\ell(G)$ be the number of distinct graph-metric lines, and let a universal pair be a pair of vertices generating a line equal to $V$; write $\textsc{up}(G)$ for the number of universal pairs. AMRZ's conjecture.

$$\ell(G)+\textsc{up}(G)\geq |V|.$$

The paper proves this inequality for metric spaces whose distances lie in $\{0,1,2\}$, while the stated conjecture for connected graphs in general remains open.