AMRZ finite-exception conjecture for graph lines and bridges

Let $G$ be a connected graph, let $\ell(G)$ be the number of distinct graph-metric lines, and let $\textsc{br}(G)$ denote the number of bridges of $G$. A pendant edge is an edge incident with a vertex of degree one. AMRZ's finite-exception conjecture. There is a finite set of graphs $\mathcal F_0$ such that every connected graph $G\notin\mathcal F_0$ either has a pendant edge or satisfies

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

The paper proves the inequality for bipartite graphs except $C_4$ and $K_{2,3}$, but the conjecture for arbitrary connected graphs remains open.