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Downward flagness after (b1,b2)-subdivision for type D_n subdiagrams

Prove that for any irreducible spherical Artin group A_S and any subset S'⊂S whose Coxeter diagram is isomorphic to the type D_n diagram (n≥3), the (b1,b2)-subdivision of the relative Artin complex Δ_{S,S'} (as defined in Definition 2.14) is a bowtie-free, downward flag poset.

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Background

This conjecture provides the D-type counterpart to the B-type link condition, now for subdivided relative complexes reflecting the D_n diagram structure. It is part of the program to reduce global contractibility and K(pi,1) to verifying specific combinatorial properties in spherical base cases.

The bowtie-free aspect is known; the downward flagness remains the missing ingredient in general beyond verified low-rank instances.

References

Conjecture. Suppose $A_S$ is an irreducible spherical Artin group. Let $S'\subset S$ such that $A_{S'}$ has Coxeter diagram isomorphic to the type $D_n$ Coxeter diagram for $n\ge 3$ (the isomorphism does not need to preserve edge labels). Let ${b_i}{i=1}{n+1}$ be vertices in $S'$ as in Figure~\ref{fig:BD} left. Then the $(b_1,b_2)$-subdivision of $\Delta{S,S'}$ (in the sense of Definition~\ref{def:subdivision}) is a bowtie free and downward flag poset.

— Cycles in spherical Deligne complexes and application to $K(π,1)$-conjecture for Artin groups (2405.12068 - Huang, 20 May 2024) in Section 2.6 (Link conditions at the base case), Conjecture 2.5