Theorem 5.1 in all dimensions at least two

Let $d$ denote the dimension of the percolation model.

Theorem 5.1 conjecture. Theorem 5.1 is true in all $d\ge2$.

The surrounding discussion indicates that the authors' control of the corrector is sufficient for the functional central limit theorem in dimensions $d\ge3$, but not for the conceptually analogous proof available in dimension two. They see no reason for a distinction between these dimensions.