Ferrara–Kay–Kramer–Martin–Reiniger–Smith–Sullivan antichain saturation asymptotic conjecture

For positive integers $k$ and $n$, let a family $\mathcal{F}\subseteq 2^{[n]}$ be $k$-antichain saturated if it contains no antichain of size $k$, but for every $X\subseteq[n]$ with $X\notin\mathcal{F}$, the family $\mathcal{F}\cup\{X\}$ contains an antichain of size $k$. Let $\operatorname{sat*}(n,k)$ be the minimum size of such a family.

Ferrara–Kay–Kramer–Martin–Reiniger–Smith–Sullivan conjecture. For $k\geq3$,

$$\operatorname{sat*}(n,k)\sim(k-1)n\quad\text{as }n\to\infty.$$

The conjecture concerns the asymptotic size of induced antichain saturation families. The paper proves exact values in several cases and later states a stronger estimate, but this asymptotic formulation is the conjecture attributed to Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan.