The multivariate Brown–Colbourn conjecture for cographic matroids

Let $G=(V,E)$ be a connected graph, let $M^*(G)$ be its cographic matroid, and let $\mathcal{I}(M^*(G))$ denote its independent sets. For channel-failure probabilities $q=\{q_e\}_{e\in E}$, define the multivariate reliability polynomial by

$$\operatorname{Rel}_{\mathcal{S}}(q)=\sum_{A\in\mathcal{S}}q^A(1-q)^{E\setminus A}.$$

Let $P_{\mathcal{S}}(x)=\sum_{A\in\mathcal{S}}x^A$ be the generating polynomial.

Multivariate Brown–Colbourn conjecture. If $|q_e|>1$ for all $e$, then

$$\operatorname{Rel}_{\mathcal{I}(M^*(G))}(q)\ne0.$$

Equivalently, if $G$ is loopless and $\operatorname{Re}x_e<-1/2$ for all $e$, then

$$P_{\mathcal{I}(M^*(G))}(x)\ne0.$$

This is a multivariate nonvanishing conjecture for all-terminal reliability of connected graphs, extending the Brown–Colbourn property from the univariate setting. The source presents it as the authors’ “Holy Grail”; no resolution is supplied here.