Griggs's chain decomposition conjecture for the Boolean algebra

Let $B_n$ be the Boolean algebra of all subsets of $[n]$, ordered by inclusion. A chain decomposition of $B_n$ has a type, namely the partition obtained by rearranging its chain sizes in weakly decreasing order. For partitions $\lambda$ and $\nu$ of $2^n$, write $\lambda\unlhd\nu$ when $\sum_{i=1}^k\lambda_i\leq\sum_{i=1}^k\nu_i$ for every $k\geq1$. Griggs's conjecture. If a partition $\lambda$ of $2^n$ satisfies $\lambda\unlhd\nu$, where $\nu$ is the partition corresponding to the symmetric chain decomposition of $B_n$, then $B_n$ has a chain decomposition of type $\lambda$. This conjecture is equivalent to the assertion that the incomparability graph of $B_n$ is nice; the paper presents it as motivation for studying Schur positivity.