Acyclic direction contractad homotopy-equivalence conjecture

Let $\Gamma$ be a graph and let $\operatorname{Conf}_{\Gamma}(\mathbb{R}^d)$ be its configuration space. Let $\operatorname{\mathsf{AD}}_d(\Gamma)$ be the poset of acyclic directions on $\Gamma$, and let $|\operatorname{\mathsf{AD}}_d(\Gamma)|$ denote its geometric realization. Consider the map

$$\psi_d\colon \operatorname{Conf}(\mathbb{R}^d)\to \operatorname{\mathsf{AD}}_d.$$

Acyclic direction contractad homotopy-equivalence conjecture. The map $\psi_d$ induces a homotopy equivalence of topological graphical collections

$$\operatorname{Conf}_{\Gamma}(\mathbb{R}^d)\simeq |\operatorname{\mathsf{AD}}_d(\Gamma)|.$$

The statement would identify the acyclic direction contractad with a combinatorial model for configuration spaces. The supplied text does not establish the claimed equivalence in this general form, so its resolution remains open.