The graph-generalized Orlik–Terao conjecture

Let $\Gamma$ be a simple connected graph with vertex set $[n]$, let $\operatorname{Aut}(\Gamma)\subset S_n$ be its automorphism group, and let $OT_\Gamma$ be the Orlik–Terao algebra of the associated graphic arrangement. Define

$$M_\Gamma:=OT_\Gamma\otimes_{R_n}\mathbb{C}.$$

Let $D_\Gamma:=H^*(\operatorname{Conf}(\Gamma,G)/G;\mathbb{C})$, where $G$ is the group used in the source, and let $\hat{\Gamma}$ be the cone over $\Gamma$; write $C_\Gamma:=H^*(\operatorname{Conf}(\Gamma,\mathbb{R}^3))$.

The graph-generalized Orlik–Terao conjecture. For any simple connected graph $\Gamma$, there exists an isomorphism

$$M_\Gamma\cong D_\Gamma$$

of graded representations of $\operatorname{Aut}(\Gamma)$. In particular, there exists an isomorphism

$$\operatorname{Res}^{\operatorname{Aut}(\hat{\Gamma})}_{\operatorname{Aut}(\Gamma)}(M_{\hat{\Gamma}})\cong C_\Gamma.$$

This extends the main conjecture from complete graphs to graphic arrangements and configuration spaces associated with arbitrary simple connected graphs. The source gives no resolution.