Upper-half dominance conjecture for chain decompositions of the Boolean lattice

Let $[n]=\{1,\dots,n\}$, let $B=[n]^{(\geq n/2)}$ be the family of subsets of $[n]$ of size at least $n/2$, 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 $2^{[n]}$, and define

$$\sigma_i'=\left\lceil\frac{\sigma_i}{2}\right\rceil.$$

Upper-half dominance conjecture. If $s_1\geq\dots\geq s_M$ is a sequence of positive integers dominated by $\sigma_1',\dots,\sigma_M'$ and

$$\sum_{i=1}^M s_i=|B|,$$

then there exists a chain decomposition $D_1,\dots,D_M$ of $B$ such that $|D_i|=s_i$ for every $i$. This is the analogous realization problem for the upper half of the Boolean lattice. The source presents it as a related conjecture to Griggs's conjecture; no resolution is supplied in the provided text.