Balogh et al.'s linear path-separating-system conjecture

Let $G$ be a graph on $n$ vertices. A path separating system of $G$ is a collection of paths in $G$ such that, for every ordered pair of distinct edges $(e,f)$, some path contains $e$ but not $f$. Balogh et al.'s conjecture. Every graph of order $n$ admits a path separating system of size $\operatorname{O}(n)$. This strengthened conjecture was confirmed in 2023, when a separating path system of size at most $19n$ was proved for every graph on $n$ vertices.