Linear growth conjecture for Berge saturation numbers

Let $k$ be fixed, and let $\mathcal{F}$ be any fixed finite family of graphs. Write $\operatorname{sat}_k(n,\text{Berge-}\mathcal{F})$ for the minimum number of edges in a $k$-uniform hypergraph on $n$ vertices that is Berge-$\mathcal{F}$-saturated. Linear growth conjecture for Berge saturation numbers. For any fixed finite family of graphs $\mathcal{F}$,

$$\operatorname{sat}_k(n,\text{Berge-}\mathcal{F})=O(n).$$

Classical graph saturation numbers are known to be linear in $n$, whereas general $k$-uniform hypergraph saturation has the upper bound $O(n^{k-1})$. The conjecture proposes linear growth for the Berge-saturation setting, but the paper does not establish it in general.