Contractibility propagation across consecutive intersections in the covering posets (Conjecture 2)

Prove that for any d ≥ 2 and any element α in the poset \overline{P}^{(d)} generated by the canonical chain of blocks used to define the arity-two component of the Θ_d-colored (d+1)-operad seq_d, if α lies in both subposets \overline{P}^{(d)}_i and \overline{P}^{(d)}_j with i < j, then α lies in each consecutive intersection subposet \overline{P}^{(d)}_{a,a+1} for all i ≤ a ≤ j−1.

Background

To prove contractibility of the arity components of seq_d, the paper covers the relevant poset by a chain of up-sets \overline{P}^{(d)}_i corresponding to canonical elements ω_i^d ∈ V^d and studies their intersections. Contractibility of the overall poset is deduced via homotopy colimit arguments, provided elements appearing in two up-sets also lie in all intermediate pairwise intersections.

Conjecture 2 formalizes this requirement and is established in the paper for d=2 and d=3. Its general validity would ensure the contractibility of P_{seq_d}(T) for all arities T and all d, completing the combinatorial step in proving the contractibility of seq_d in both topological and dg condensations.