The quasi-center conjecture for zigzag 6-cycles in type $D_n$ Artin complexes

Let $\Lambda$ be the Dynkin diagram of type $D_n$, and let $\Delta_\Lambda$ be its associated Artin complex. Order the vertex types by $\{\hat\delta_1,\hat\delta_2\}<\hat\delta_3<\hat\delta_4<\cdots<\hat\delta_n$, with $\hat\delta_1$ and $\hat\delta_2$ incomparable. A 6-cycle is admissible if consecutive vertex types are comparable; a vertex is a local maximum or minimum according to this order, and a zigzag 6-cycle is an admissible 6-cycle whose vertices alternate between local maxima and local minima. Quasi-center conjecture. Every zigzag 6-cycle $\omega$ in $\Delta_\Lambda$ has a quasi-center adjacent to each local maximum vertex of $\omega$. This conjecture is formulated as part of the proposed approach to the $K(\pi,1)$ conjecture and is known in the source for type $D_n$ Artin groups with $n=3,4$, while the general case remains open.