Clustering distribution on the two-dimensional integer lattice

Let $D\subset\mathbb R^d$ be open, bounded, and convex, and let $\mathcal W$ be a renormalization class on $\overline D$. Assume that the asymptotic fixed-point equation has a unique solution $w^*\in\mathcal W$. Let $\sigma$ be a continuous root of $w\in\mathcal W$, and let $(\mathbf x_\xi)_{\xi\in\mathbb Z^2}$ solve

$$d\mathbf x_\xi(t)=\sum_{\eta:\,|\eta-\xi|=1}(\mathbf x_\eta(t)-\mathbf x_\xi(t))\,dt+\sigma(\mathbf x_\xi(t))\,dB_\xi(t),$$

with $\mathbf x_\xi(0)=\theta\in\overline D$. Clustering distribution on the two-dimensional integer lattice. As $t\to\infty$, the law of the origin converges to the limiting distribution of the diffusion $I^0$ with generator $\sum_{i,j}w^*_{ij}(y)\frac{\partial^2}{\partial y_i\partial y_j}$, started at $\theta$:

$$\mathcal L(\mathbf x_0(t))\Longrightarrow P[I^0_\infty\mid I^0_0=\theta].$$

This conjecture relies on uniqueness of the asymptotic fixed point and concerns clustering for the interacting diffusion on $\mathbb Z^2$; the source gives it as a further conjecture rather than a proved result.