Monotone diameter-2 path-system conjecture

Let $f:\binom{[n]}{2}\to [n]$ be the unique résumé of a diameter-$2$ path system $\mathcal{P}$. The system $\mathcal{P}$ is monotone if $f(i,j)\le f(i,k)$ whenever $\{i,j\},\{i,k\}\in\operatorname{dom} f$ and $j\le k$. Monotone path-system conjecture. Every monotone consistent path system of diameter $2$ is strictly metric. This would extend the classes of monotone path systems shown in the paper to be strictly metric, and concerns the structure of the metric cone and its path systems. Its resolution is not given here.