Griggs's dominance conjecture for chain decompositions of the Boolean lattice

Let $[n]=\{1,\dots,n\}$, let $2^{[n]}$ be the Boolean lattice, and set

$$M=\binom{n}{\lfloor n/2\rfloor}.$$

Let $\sigma_1\geq\dots\geq\sigma_M$ be the sizes of the chains in a symmetric chain decomposition of $2^{[n]}$. A sequence $s_1,\dots,s_M$ is dominated by $\sigma_1,\dots,\sigma_M$ when

$$\sum_{i=1}^k s_i\leq\sum_{i=1}^k\sigma_i\quad (k=1,\dots,M).$$

Griggs's conjecture. If $s_1\geq\dots\geq s_M$ is a sequence of positive integers dominated by $\sigma_1,\dots,\sigma_M$ and

$$\sum_{i=1}^M s_i=2^n,$$

then there exists a chain decomposition $D_1,\dots,D_M$ of $2^{[n]}$ such that $|D_i|=s_i$ for every $i$. The conjecture proposes that every admissible chain-size sequence below the symmetric-chain profile can be realized. The source attributes it to Griggs; it notes that the uniform chain decomposition conjecture is a special subcase and may be the most challenging one.