Conjecture on uniform cutset sizes in the truncated Boolean lattice

Let $g_n(m,l)$ denote the minimum size of a cutset in the truncated Boolean lattice with parameters $n$, $m$, and $l$. For $n\gg m$, the following assertions are conjectured.

Uniform cutset conjecture.

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

for every $l=2m,2m+1,\dots,n-m-1$, and

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

These claims describe the observed stabilization of $g_n(m,l)$ across the middle range of levels and its separate value at the endpoint $l=n-m$. The condition $n\gg m$ is informal in the source, and the conjectural formulas are not established there.