The linear-growth conjecture for complete posets

Let $K_{n_1,\dots,n_k}$ denote the complete poset with consecutive layers of sizes $n_1,\dots,n_k$, where $n_1,\dots,n_k$ are positive integers and at least one $n_i$ is not equal to $1$. Let $\operatorname{sat}^*(n,K_{n_1,\dots,n_k})$ denote its induced saturation number. Complete-poset linear-growth conjecture.

$$\operatorname{sat}^*(n,K_{n_1,\dots,n_k})=\Theta(n).$$

The paper notes that it cannot obtain linear upper bounds for complete posets having at least two consecutive singleton layers, and proposes this statement as a conjectural resolution for the indicated class.