The chain-is-best conjecture for poset saturation

Let $P$ be a poset with $k$ elements, let $C_k$ denote the chain on $k$ elements, and let ${\rm sat}(n,P)$ be the minimum size of a maximal $P$-free subposet of the Boolean lattice $B_n$.

Chain-is-best conjecture. For any $k$-element poset $P$, we have

$${\rm sat}(n,P)\le {\rm sat}(n,C_k).$$

The conjecture asks whether the chain is the hardest $k$-element poset to saturate. The paper has already established the general upper bound ${\rm sat}(n,P)\le 2^{k-2}$, while the known lower bound for chains shows that exponential growth in $k$ is possible; the comparison with ${\rm sat}(n,C_k)$ remains open.