Gaussian free field scaling conjecture for the percolation corrector

Let $\chi(x,\omega)$ be the corrector on the infinite percolation cluster, and let $\Delta$ be the Dirichlet Laplacian on $\mathbb R^d$. Write $\mathbf 1$ for the $d$-dimensional unit matrix. For $x\in\mathbb R^d$ and $[?]$ let $\lfloor x/\epsilon\rfloor$ denote the coordinatewise lattice approximation.

Corrector Gaussian free field conjecture. Let $d\ge1$. Then the law of

$$x\mapsto\epsilon^{\frac{2-d}{2}}\chi\left(\left\lfloor\frac{x}{\epsilon}\right\rfloor\right)$$

on compact subsets of $\mathbb R^d$ converges weakly, as $\epsilon\downarrow0$, to the Gaussian Free Field, namely a multivariate Gaussian field with covariance proportional to $\Delta^{-1}\mathbf 1$.

This conjecture proposes a scaling limit for the corrector in all dimensions, complementing the high-dimensional tightness conjecture. The source describes it as a substantially less certain guess; it also notes that the corresponding one-dimensional statement with conductances bounded away from zero is a theorem.