The IIP conjecture beyond moment conditions and ergodicity

Let $d\ge1$, let $\Omega^\star$ and $\Omega_{p,q}$ denote the sets of environments defined in the paper, and let $\mathrm{IIP}$ denote the invariance invariance principle established there. The IIP conjecture. In all $d\ge1$, an IIP holds for all $\omega\in\Omega^\star\cap\bigcup_{p,q>1}\Omega_{p,q}$. This would show that the moment conditions used in the present method are artifacts rather than intrinsic requirements, extending the deterministic homogenization result beyond the currently treated setting.