Linear induced saturation conjecture for disjoint unions of chains

Let $C_k$ denote the chain poset with $k$ elements, and let $\bigoplus_{j=1}^m C_{i_j}$ be the disjoint union of chains with $i_1\ge i_2\ge\dots\ge i_m$. For a poset $P$, write $\operatorname{sat}^*(n,P)$ for the minimum size of an induced $P$-saturated family in $2^{[n]}$. Linear induced saturation conjecture. For any $i_1\ge i_2\ge\dots\ge i_m$, there exists $C=C(i_1,i_2,\dots,i_m)$ such that

$$\operatorname{sat}^*(n,\bigoplus_{j=1}^m C_{i_j})\le C\cdot n.$$

At the time of the paper, it was unknown whether any poset has superlinear induced saturation number. If true, this conjecture is best possible because $\operatorname{sat}^*(n,2C_2)=\Theta(n)$ is known.