Parabolic aligned elements and flips of the parabolic cluster complex

Let $(W,S)$ be a finite Coxeter system, let $J\subseteq S$, and let $c\in W$ be a Coxeter element. The $(W^J,c)$-cluster complex is the subword complex

$$\operatorname{SW}\bigl(\mathbf{c w_{\circ}(c)},w_{\circ}^{J}\bigr),$$

and its facets carry the flip poset structure: a flip $F\to F'$ replaces positions $i<j$ when $F-\{i\}=F'-\{j\}$. Parabolic flip conjecture. For any finite Coxeter system $(W,S)$, any $J\subseteq S$, and any Coxeter element $c\in W$, the restriction of the weak order to $(W^{J},c)$-aligned elements is isomorphic to the flip poset of $\operatorname{SW}(W^{J},c)$. The conjecture would identify the weak-order model of parabolic aligned elements with the facet-flip model of the corresponding subword complex. It is disproved: the source states that the assertion does not hold in general for parabolic quotients and Coxeter elements.