Saturation of $k$-chains in the Boolean lattice

Given a set $X$, a collection $\mathcal{F} \subset \mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this probability. Gerbner et al. proved that the smallest saturated $k$-Sperner system contains at least $2^{k/2-1}$ elements, and later, Morrison, Noel, and Scott showed that the smallest such set contains no more than $2^{0.976723k}$ elements. We improve both the upper and lower bounds, showing that the size of the smallest saturated $k$-Sperner system lies between $\sqrt{k}2^{k/2}$ and $2^{0.961471k}$.