The two-dimensional martingale-field convergence conjecture for the voter model

Let $T>0$, let $\nu_p$ be the product Bernoulli measure of density $p$, and let $\hat{\mathcal{Y}}^N$ and $\hat{\mathcal{Y}}$ be the discrete and limiting random fields defined in the paper on test functions over $[0,T]\times\mathbb{R}^2$. For $H\in C_c([0,T]\times\mathbb{R}^2)$, write $\hat{\mathcal{Y}}^N(H)$ and $\hat{\mathcal{Y}}(H)$ for their evaluations at $H$. Two-dimensional martingale-field convergence conjecture. If $\eta_0$ is distributed according to $\nu_p$, then for every $H\in C_c([0,T]\times\mathbb{R}^2)$, $\hat{\mathcal{Y}}^N(H)$ converges weakly to $\hat{\mathcal{Y}}(H)$ as $N\to+\infty$, and

$$\lim_{N\to+\infty}\mathbb{E}_{\nu_p}\left(\left(\hat{\mathcal{Y}}^N(H)\right)^2\right)=\mathbb{E}\left(\left(\hat{\mathcal{Y}}(H)\right)^2\right).$$

This is presented as the two-dimensional analogue of the corresponding martingale-field convergence lemma and would imply the two-dimensional occupation-time conjecture. Its status is open in the source.