Thermodynamic-limit conjecture for the random-cluster partition function

Let $G_n$ be a graph sequence that converges locally to a Galton--Watson tree. Let $Z_{G_n}(q,w,B)$ be the random-cluster partition function with parameters $q,w,B$, and define the pressure per particle by

$$\varphi_n(q,w,B)=\frac{1}{n}\log Z_{G_n}(q,w,B).$$

Thermodynamic limit of the random-cluster partition function. As $n\rightarrow\infty$,

$$\varphi_n(q,w,B)\longrightarrow\varphi(q,w,B),$$

where $\varphi(q,w,B)$ is defined in the cited equation in the source. This is stated as a conjecture concerning convergence of the pressure for random-cluster models on graphs converging locally to a Galton--Watson tree; the supplied text gives no resolution.