Ivan's complete bipartite poset saturation conjecture

For $s\geq 2$, let $\mathcal{K}_{s,t}$ be the complete bipartite poset with $s$ upper-layer and $t$ lower-layer pairwise incomparable vertices, every upper-layer vertex larger than every lower-layer vertex. Let $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ denote the smallest size of an induced $\mathcal{K}_{s,t}$-saturated family in $2^{[n]}$. Ivan's conjecture. For all $s\geq 2$,

$$\mathrm{sat}^{*}(n,\mathcal{K}_{s,2})=\Theta(n^{s})\quad\text{and}\quad\mathrm{sat}^{*}(n,\mathcal{K}_{s,s})=\Theta(n^{2s-2}).$$

The source notes that the corresponding upper bounds were proved, while the conjecture asks for matching lower bounds and strengthens the earlier $\mathcal{K}_{2,2}$ conjecture.