Bounded modified saturation for an isolated minimal edge

Let $G$) be an edge-ordered graph and let $e_0$ be an isolated minimal edge of $G$. Write $F=e_0+G$. The isolated-edge saturation conjecture.

$$sat_m(n,F)=O(1).$$

The authors present this as a candidate explaining why no edge-ordered graph with linear $sat_m$ saturation and isolated minimal edge is known. The analogous statement for $sat_e$ is false.