Linear-order conjecture for complete bipartite poset saturation

Let $s\geq t\geq 2$ be fixed, and 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 $2^{[n]}$. The linear-order conjecture.

$$\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})=\Theta(n)$$

for all fixed $s\geq t\geq 2$. The conjecture proposes that the upper bound proved earlier in the paper is tight up to a multiplicative constant, extending the determined linear order for the $t=2$ case.