Linear modified saturation under disjoint neighborhoods

Let $G$ be an edge-ordered graph, and let $ab$ be its minimal edge such that $\deg(a)\geq 2$. Assume that the neighborhoods of its endpoints are disjoint:

$$N_G(a)\cap N_G(b)=\emptyset.$$

Disjoint-neighborhood saturation conjecture. Under these hypotheses,

$$sat_m(n,G)=\Theta(n).$$

Theorem-level results in the paper establish linear semisaturation under the same neighborhood condition, but the corresponding assertion for the saturation function remains open; the authors report no counterexample.