Haettel's bowtie-free and upward-flag conjecture for type $D_n$ subdivisions

Let $d6f2$ be a Coxeter diagram of type $D_n$ with $n\geq 3$. Construct the $(s_n,s_{n-1})$-subdivision $d6f2'_d6f2$ of the simplicial complex $d6f2_d6f2$ by subdividing each edge connecting a vertex of type $\hat s_n$ to one of type $\hat s_{n-1}$, assigning the new midpoint type $m$, and cutting each top-dimensional simplex along the codimension-one simplex spanned by the vertices of type $m$ and $\{\hat s_i\}_{i=1}^{n-2}$. Define $t$ on the vertices of $d6f2'_d6f2$ by sending types $\hat s_i$ to $i$ for $1\leq i\leq n-2$, type $m$ to $n-1$, and types $\hat s_n,\hat s_{n-1}$ to $n$. For adjacent vertices $x,y$, set $x<y$ when $t(x)<t(y)$. Haettel's conjecture. The relation $<$ makes $((d6f2'_d6f2)^0,<)$ a poset that is bowtie free and upward flag. This conjecture predicts that the type $D_n$ subdivision shares a key combinatorial property with the Artin complex of type $B_n$; Haettel's original motivation was to obtain an alternative proof that the Artin group of type $\widetilde D_n$ satisfies the $K(\pi,1)$-conjecture.