Near-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. Near-linear saturation conjecture. For every edge-ordered graph $G$,

$$sat_e(n,G)=O\left(n^{1+o(1)}\right).$$

The paper identifies this as the main general upper-bound problem and notes that the same bound would imply the corresponding bound for $sat_m$.