Nonvanishing conjecture for graph amplitude functions

Let $G$ be a non-oriented finite simple graph, let $C_G(\mathbb{R}^3)$ be its configuration space, and let $\mathcal{A}_G$ denote the associated $G$-amplitude function. Graph-amplitude nonvanishing conjecture. The function $\mathcal{A}_G$ is nowhere vanishing on $C_G(\mathbb{R}^3)$. This is the first graph-theoretic analogue of the Atiyah conjecture introduced in the paper; no resolution is given in the supplied text.