The minimum-width conjecture for cutsets in the truncated Boolean lattice

Let $[n]=\{1,2,\ldots,n\}$, let $2^{[n]}$ be the Boolean lattice ordered by inclusion, and let

$${\cal B}_n(m,l)=\bigcup_{i=m}^{l}\binom{[n]}{i},$$

where $0\leq m\leq l\leq n-m$. A cutset is a collection ${\cal C}\subseteq{\cal B}_n(m,l)$ meeting every maximal saturated chain, and $h_n(m,l)$ is the minimum width of such a cutset. Set $c=l-m+1$ and

$$\Delta_n(k)=\binom{n}{k}-\binom{n}{k-1}.$$

Minimum-width conjecture. For $n\gg m$,

$$h_n(m,l)=\sum_{j\geq0}\Delta_n(m-jc)=\Delta_n(m)+\Delta_n(m-c)+\Delta_n(m-2c)+\cdots.$$

This conjecture seeks the minimum possible largest antichain in a cutset of the truncated Boolean lattice, generalizing the known result $h_n(1,n-1)=n-1$. The source gives no resolution, so its status remains open.