Botler–Naia conjecture on linear-size subdivision separating systems

Let $H$ be a graph with at least one edge. A subdivision of $H$ is obtained by replacing edges of $H$ with pairwise internally vertex-disjoint paths; let $\operatorname{Sub}(H)$ be the class consisting of $K_2$ and all subdivisions of $H$. A separating $\operatorname{Sub}(H)$-system is a family of subgraphs from this class that separates the edges of the host graph. Botler–Naia conjecture. For every graph $H$ with at least one edge, there is a constant $C_H>0$ such that every $n$-vertex graph admits a separating $\operatorname{Sub}(H)$-system of size at most $C_Hn$. Botler and Naia proved the corresponding assertion for $H=K_3$ and $H=K_4$, with bounds $41n$ and $82n$, respectively; the conjecture asks for the result for every graph $H$ with an edge.