The equality conjecture for level-width and cutset-width in the truncated Boolean lattice

Let $[n]=\{1,2,\ldots,n\}$ and let ${\cal B}_n(m,l)$ be the truncated Boolean lattice consisting of all subsets of $[n]$ of sizes from $m$ through $l$. A cutset is a collection meeting every maximal saturated chain. Let $h_n(m,l)$ be the minimum width of a cutset, and let $g_n(m,l)$ be the smallest $k$ for which there is a cutset containing at most $k$ nodes at each level. Equality conjecture. If $0\leq m\leq l\leq n-m-1$ and $n\gg m$, then

$$g_n(m,l)=h_n(m,l).$$

In addition,

$$g_n(m,n-m)=\binom{n-1}{m}-\binom{n-1}{m-1}.$$

The inequality $h_n(m,l)\geq g_n(m,l)$ is immediate from the definitions, while the conjecture predicts equality in the stated range and gives an additional boundary formula. The source gives no resolution, so its status remains open.