Saturating Sperner families

A family $\text{cF}\subseteq 2^{[n]}$ saturates the monotone decreasing property $\text{cP}$ if $\text{cF}$ satisfies $\text{cP}$ and one cannot add any set to $\text{cF}$ such that property $\text{cP}$ is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the $k$-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of $l$-sets and $(l+1)$-sets.