The two-dimensional sample path central limit conjecture for voter-model occupation time

Let $T>0$, let $u_p$ be the product Bernoulli measure of density $p$, let $u_p$ govern the initial voter-model configuration $\eta_0$, and let $\eta_s(O)$ denote the occupation state at the origin at time $s$. Let $\{\vartheta_t\}_{t\geq 0}$ be the centered Gaussian process with continuous sample paths and covariance, for $0\leq s\leq t$,

$$\operatorname{Cov}(\vartheta_t,\vartheta_s)=\frac{(t+s)^2}{4}\log(t+s)+\frac{(t-s)^2}{4}\log(t-s)-\frac{s^2\log s}{2}-\frac{t^2\log t}{2}.$$

Two-dimensional occupation-time central limit conjecture. If $d=2$ and $\eta_0$ is distributed according to $\nu_p$, then

$$\left\{\left(\frac{N}{\sqrt{\log N}}\right)^{-1}\int_0^{tN}\left(\eta_s(O)-p\right)\,ds:0\leq t\leq T\right\}_{N\geq 1}$$

converges weakly, with respect to the Skorohod topology of $C[0,T]$, to $\{C_2\vartheta_t\}_{0\leq t\leq T}$ as $N\to+\infty$, where $C_2=\sqrt{2p(1-p)}$. This is the conjectured two-dimensional analogue of the proved sample path central limit theorem in dimensions $d\geq 3$; the logarithmic normalization and the nonstandard Gaussian limit arise from the recurrent two-dimensional random walk, and the claim remains open in the source.