Zero-freeness transfer to a neighborhood of the Ising points

Fix a nonzero parameter $\lambda$, and let $Z^{\sigma_\Lambda}_{G}(\beta)$ denote the partition function of the Ising model with boundary condition $\sigma_\Lambda$. Suppose it is zero-free on a complex region $U$. Ising zero-freeness transfer conjecture. There exists a complex neighborhood $U'$ of $1$ or $-1$ such that $U\subseteq U'$ and $Z^{\sigma_\Lambda}_{G}(\beta)$ is zero-free on $U'$. This specializes the proposed zero-freeness principle to the Ising model, whose relevant special points are $1$ and $-1$. The source gives no evidence of resolution, so the conjecture remains open.