Asymptotic sharpness of the layered lower bound for Boolean-lattice chain covers

Let $B_n$ be the Boolean lattice of subsets of an $n$-element set, and let $N(n,r)$ denote the minimum number of maximal chains needed to cover all strict $r$-term chains in $B_n$. For an admissible gap vector $(a_0,\dots,a_r)$, meaning $a_0,a_r e 0$ or $a_0,a_r\ge 0$ and $a_i\ge 1$ for $1\le i\le r-1$, with $a_0+\cdots+a_r=n$, define

$$M(n,r):=\max_{\mathbf a}\binom{n}{a_0,a_1,\dots,a_r}.$$

Asymptotic sharpness conjecture. For every fixed $r\ge 1$,

$$N(n,r)=(1+o(1))M(n,r).$$

The quantity $M(n,r)$ is a layered lower bound obtained by fixing the gap vector of the covered chains. The conjecture is disproved: the exact equality version is false in general, as shown by the paper's discussion of small values.