Kleitman's conjecture on minimizing the number of k-chains

Let $[n]=\{1,\ldots,n\}$ and let $\mathcal{P}(n)=2^{[n]}$ be the Boolean lattice. For integers $n,M>0$, a family $\mathcal{F}\subseteq\mathcal{P}(n)$ of size $M$ is centered if its sets are chosen from the layers whose sizes are as close to $n/2$ as possible, filling the larger set-size first when two layers are equally distant from $n/2$. A $k$-chain is a sequence of $k$ sets $F_1\subsetneq\cdots\subsetneq F_k$.

KleItman's conjecture. Let $n,M>0$ and $k\geq 2$ be integers. Among families $\mathcal{F}\subseteq\mathcal{P}(n)$ of size $M$, the number of $k$-chains in $\mathcal{F}$ is minimized by a centered family.

Kleitman proved the assertion for $k=2$, while the conjecture for general $k$ is the main open problem addressed by the paper. The results establish it in a wide range of parameters, but not in full generality.