McConville–Sagan–Smyth chain decomposition conjecture for fence lattices

Let $F(\beta)$ be a fence and let $L(\beta)$ be its distributive lattice of lower order ideals, with rank sequence governed by the alternatives in Theorem heavy. A saturated chain decomposition is an SCD if every chain has center $n/2$, a TCD if every chain has center $n/2$ or $(n+1)/2$, and a BCD if every chain has center $n/2$ or $(n-1)/2$, where $n$ is the maximum rank. McConville–Sagan–Smyth's chain decomposition conjecture. The lattice $L(\beta)$ admits an SCD, BCD, or TCD consistent with Theorem heavy. Such decompositions would explain the corresponding symmetry, bottom-interlacing, or top-interlacing rank-sequence behavior. The source reports proofs for fences with at most three parts and for fences of the form $(k,1,k,1,\ldots,k,1,l)$ with $1\le l\le k$, while the general conjecture remains open.