Almost-linear saturation conjecture for edge-ordered graphs

Let $G$ be an edge-ordered graph, and let $sat_e(n,G)$ denote its edge-ordered saturation function. Almost-linear saturation conjecture. For every edge-ordered graph $G$,

$$sat_e(n,G)=O(n\log n).$$

This is stated as a stronger conjecture than the paper's general $O\left(n^{1+o(1)}\right)$ upper-bound conjecture. The authors regard it as plausible by analogy with the known upper bound for semisaturation.