Rainbow saturation conjecture for uniform hypergraphs

Let $H$ be a $k$-uniform hypergraph, and let ${\operatorname{sat}^\star}_k(n,H)$ denote the minimum number of hyperedges in an $n$-vertex rainbow $H$-saturated $k$-uniform hypergraph.

Rainbow hypergraph saturation conjecture. For every $k$-uniform hypergraph $H$, we have

$${\operatorname{sat}^\star}_k(n,H)=O(n^{k-1}).$$

This conjecture is the rainbow analogue of Pikhurko's bound for ordinary saturation in uniform hypergraphs. The proof method developed for graphs extends with minor modification, but the authors note that an appropriate hypergraph analogue of their tree result is unavailable, leaving the conjecture open.