Higher-uniformity Berge saturation conjecture

Let $3\leq r<k$ be integers and let $F^{(r)}$ be an $r$-uniform hypergraph. A $k$-uniform hypergraph $H$ is $\text{Berge}_k\text{-}F^{(r)}$ if there is a bijection

$$\phi:E(F^{(r)})\to E(H)$$

such that $e\subseteq\phi(e)$ for every $e\in E(F^{(r)})$. Let $\operatorname{sat}_k(n,\text{Berge-}F^{(r)})$ denote the minimum number of hyperedges in a $k$-uniform hypergraph on $n$ vertices that is free of this configuration but becomes non-free after adding any $k$-edge.

Higher-uniformity Berge saturation conjecture. For every $3\leq r<k$ and every $r$-uniform hypergraph $F^{(r)}$,

$$\operatorname{sat}_k(n,\text{Berge-}F^{(r)})=O(n^{r-1}).$$

This generalizes the earlier conjecture that Berge saturation numbers are linear when the forbidden object is a graph, corresponding to $r=2$. The conjecture predicts the natural $r-1$ power growth for higher-uniformity forbidden hypergraphs; its general case remains open.