Hardness conjecture from zeros of the complex-valued Ising partition function

Let $\Delta$ be an integer with $\Delta \ge 3$ and let $\beta\in \mathbb{C}_{\mathbb{A}}$ with $\beta \not \in \mathbb{R}\cup \{i, -i\}$. Let $G$ be a graph with maximum degree at most $\Delta$, and let $Z_{\mathrm{Ising}}(G; \beta)$ denote its Ising partition function. The approximation problems $\textsc{IsingNorm}(\beta, \Delta, 1.01)$ and $\textsc{IsingArg}(\beta, \Delta, \pi/3)$ concern, respectively, approximating the norm and argument of this partition function. Hardness conjecture from zeros. If there is a graph $G$ with maximum degree at most $\Delta$ such that

$$Z_{\mathrm{Ising}}(G; \beta) = 0,$$

then the problems

$$\textsc{IsingNorm}(\beta, \Delta, 1.01) \quad\text{and}\quad \textsc{IsingArg}(\beta, \Delta, \pi/3)$$

are $\mathrm{\#P}$-hard. The surrounding discussion explains that a zero of the partition function can typically be used to implement the edge interaction $-1$, which yields hardness through the preceding lemma; the conjecture concerns the case where terminal-degree conditions are not established.