Rank-unimodality conjecture for composition and layered-permutation downsets

Let $w$ be a composition, ordered by the subword order, and let its principal downset consist of all compositions contained in $w$. A composition is associated with a layered permutation by replacing its parts with decreasing blocks. Composition and layered-permutation conjecture. The principal downset of every composition is rank-unimodal in the subword order. Hence, the principal downset of every layered permutation is rank-unimodal. The claim is stated as an intriguing open problem; only compositions with monotone parts are covered by the result discussed immediately before it.