Linear-size separating path systems conjecture

Let $G$ be a graph on $n$ vertices. A separating path system is a collection of paths in $G$ that separates the edges of $G$, either weakly or strongly. Linear-size separating path systems conjecture. There exists a constant $c$ such that every graph on $n$ vertices has a weakly- or strongly-separating path system of size at most $cn$. This conjecture concerns Katona's question about the maximum minimum size of separating path systems over $n$-vertex graphs. The source gives no resolution, so the conjecture remains open; the strongly separating version is slightly stronger than the weak version.