Upward flagness for linear subdiagrams with a heavy edge (type B-style conjecture)
Prove that for any irreducible spherical Artin group A_Λ and any linear subdiagram Λ'⊂Λ with consecutive vertices {s_i}_{i=1}^n such that the edge between s_{n−1} and s_n has label at least 4, the vertex set of the relative Artin complex Δ_{Λ,Λ'} endowed with the order induced by s_1<⋯<s_n forms a bowtie-free, upward flag poset.
References
Conjecture. Suppose $A_S$ is an irreducible spherical Artin group with Coxeter diagram $\Lambda$. Let $\Lambda'$ be a linear subdiagram of $\Lambda$ with consecutive vertices ${s_i}{i=1}n$ such that the edge between $s{n-1}$ and $s_n$ has label $\ge 4$. Then the vertex set of $\Delta_{\Lambda,\Lambda'}$ equipped with the relation induced from $s_1<s_2<\cdots<s_n$ (as in Definition~\ref{def:order}) is a bowtie free and upward flag poset.