Zero-freeness transfer to a neighborhood of zero for 2-spin systems

Fix parameters $\beta$ and $\gamma$, not both zero, and let $Z^{\sigma_\Lambda}_{G}(\lambda)$ denote the partition function of the 2-spin system with boundary condition $\sigma_\Lambda$. Suppose it is zero-free on a complex region $U$. Zero-freeness transfer conjecture. There exists a complex neighborhood $U'$ of $0$ such that $U\subseteq U'$ and $Z^{\sigma_\Lambda}_{G}(\lambda)$ is zero-free on $U'$. The paper asks whether all zero-free regions for 2-spin systems must arise in complex neighborhoods of $\lambda=0$ or, after a spin switch, from the other special parameter setting. This remains open in the source.