The inverse-square variance conjecture for left-right resistance on square grids

Consider the graph $\mathbb{Z}^2\cap[0,n]\times[0,n]$, with source $A_n=\\{0\\}\times[0,n]$ and sink $Z_n=\\{n\\}\times[0,n]$. Let $\mathcal{R}_r(A_n\leftrightarrow Z_n)$ denote the effective resistance between these sets for i.i.d. edge resistances that are bounded away from $0$ and infinity.

Inverse-square variance conjecture.

$$\operatorname{Var}(\mathcal{R}_r(A_n\leftrightarrow Z_n))=O\left(\frac{1}{n^2}\right).$$

The conjecture concerns concentration of the left-right effective resistance in a square grid. The source says that a Poincare-type inequality suggests this rate, but does not establish it; the almost-sure limiting resistance mentioned nearby is described as known under an ellipticity condition.