Saturation for Small Antichains

For a given positive integer $k$ we say that family of subsets $[n]$ is $k$-antichain saturated if it does not contain pairwise incomparable sets, but whenever add to new set, do find such sets. The size the smallest denoted by $\text{sat}^*(n, \mathcal A_{k})$. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured A_{k})=(k-1)n(1+o(1))$, proved this for $k\leq 4$. In paper prove conjecture $k=5$ $k=6$. Moreover, give exact value A_5)$ A_6)$. We also some open problems inspired our analysis.