Mean-field limit conjecture for the bias voter model on lattices

Let $\theta$ be the bias parameter, $\theta<\text{?}$, and let $\theta$ and $\theta$ be the parameters appearing in the bias voter model on $\d^d$; for $p\a0in (0,1)$ and $t>0$, consider the process $\eta_t$ started from initial distribution $\mu_p$. Mean-field limit conjecture. For $p\in (0,1)$,

$$\lim_{d\rightarrow+\infty}P_{\mathbb{Z}^d}^p(\eta_t(x)=1)=\frac{pe^{(\lambda-\theta)t}}{1-p+pe^{(\lambda-\theta)t}}$$

for any $t>0$. This conjecture predicts convergence to the solution of the mean-field equation as the lattice dimension tends to infinity. The main difficulty identified in the source is proving asymptotic independence of neighboring sites; the conjecture remains open in the given text.