Pattern-conditioned symmetric chain decomposition conjecture for Young's lattice boxes

Let $L(w,h)$ be the poset of partitions in a box of height $h$ and width $w$, ordered componentwise. A saturated chain in $L(w,h)$ has an edge labelling recording the columns in which boxes are added. Say that a chain is described by a pattern if its edge labelling consists first of entries with no $1$s, followed by repeated permutations of all column labels. A symmetric chain decomposition satisfies the restriction, extension, and pattern conditions when every chain has the corresponding properties described in the source. Pattern-conditioned decomposition conjecture. Symmetric chain decompositions satisfying the restriction, extension, and pattern conditions exist for all $L(w,h)$.

The source constructs such decompositions for $w=3,4$ and describes the case $w=2$, providing evidence for the conjecture. The assertion for general $w$ remains open in the supplied text.