Lexicographic chain decomposition conjecture for fence lattices

Let $F(\beta)$ be a fence and let $L(\beta)$ be its distributive lattice of lower order ideals. For a linear extension of $F(\beta)$, form the corresponding lexicographic chain decomposition (LCD) of $L(\beta)$ by repeatedly choosing the lexicographically smallest available ideal of minimum rank and extending it through lexicographically smallest available covers. Lexicographic chain decomposition conjecture. For any $\beta$, there is a linear extension of $F(\beta)$ whose corresponding LCD is an SCD, BCD, or TCD of $L(\beta)$ consistent with Theorem heavy. If true, this would provide a constructive realization of the chain-decomposition alternative for every fence; the source presents it as an unresolved conjectural inductive phenomenon.