Thermodynamic-limit conjecture for the extended Ising model

Let $G_n$ be a graph sequence that is locally tree-like. Let $Z_{G_n}^{\scriptscriptstyle \rm eIsing}(\beta,k,h)$ denote the partition function of the extended Ising model with parameters $\beta,k,h$ defined as in the model's definition. Write

$$\varphi_n^{\scriptscriptstyle \rm eIsing}(\beta,k,h)=\frac{1}{n}\log Z_{G_n}^{\scriptscriptstyle \rm eIsing}(\beta,k,h).$$

Pressure per particle of the extended Ising model. There exists $\varphi^{\scriptscriptstyle \rm eIsing}(\beta,k,h)$ such that

$$\varphi_n^{\scriptscriptstyle \rm eIsing}(\beta,k,h)\longrightarrow \varphi^{\scriptscriptstyle \rm eIsing}(\beta,k,h).$$

This conjectures existence of the thermodynamic limit for the extended Ising model even when the external fields do not all have the same sign. The supplied text presents it as an open conjecture and gives no resolution.