The $K_{2,k}$ induced saturation conjecture

Let $[n]=\{1,2,\ldots,n\}$, let $K_{2,k}$ denote the corresponding poset, and let $\operatorname{sat}^*(n,K_{2,k})$ be the smallest size of a $K_{2,k}$-saturated family of subsets of $[n]$. The $K_{2,k}$ induced saturation conjecture.

$$\operatorname{sat}^*(n,K_{2,k})=\Theta(n^k).$$

The conjecture is motivated by a construction using the singletons and a saturated chain, with additional sets concentrated in the lower levels of the power set. The source does not provide a proof of the matching asymptotic lower and upper bounds.