Downward-flag conjecture for $D_n$ subdivisions of relative Artin complexes

Let $A_S$ be an irreducible spherical Artin group. Let $S'\subset S$ be such that $A_{S'}$ has Dynkin diagram isomorphic to type $D_n$ for $n\geq 3$, without requiring the isomorphism to preserve edge labels. Let $\{b_i\}_{i=1}^{n+1}$ be the vertices specified in the source's Figure~ left, and form the $(b_1,b_2)$-subdivision of $\Delta_{S,S'}$. $D_n$ subdivision conjecture. This subdivision is a bowtie-free and downward-flag poset. The bowtie-free assertion and several special cases of the downward-flag assertion are known, but the remaining cases are open.