High-dimensional tightness conjecture for the percolation corrector

Let $\chi(x,\omega)$ be the corrector on the infinite percolation cluster $\mathscr C_\infty(\omega)$, and let $\mathbb P_0$ denote the relevant probability measure conditioned on the origin belonging to the infinite cluster. For sufficiently large dimension $d$, consider the corrector evaluated at $x\in\mathbb Z^d$.

High-dimensional corrector tightness conjecture. Let $d\gg1$. Then for each $\epsilon>0$ there exists $K<\infty$ such that

$$\mathbb P_0\bigl(|\chi(x,\omega)|>K\mid x\in\mathscr C_\infty\bigr)<\epsilon$$

for all $x\in\mathbb Z^d$.

The claim says that the corrector remains tight uniformly over spatial locations in sufficiently high dimensions. The authors suggest that Barlow's heat-kernel estimates might prove it, and note that it would explain the relation with their low-dimensional scaling conjecture.