The disjoint-edge bound for geometric graphs

Let $m$ be a non-negative integer and $G$ be a geometric graph such that $|DJ(uv)|\leq m$ for any edge $uv\in E(G)$. The disjoint-edge bound. Then

$$|E(G)|\leq \frac{\sqrt{1+8m}+3}{4}\cdot |V(G)|.$$

The conjecture proposes a sharp-looking linear bound on the number of edges in terms of the maximum number of edges disjoint from an individual edge. The source gives no resolution, so the claim remains open.