The bounded-variance conjecture for effective resistance on the square lattice

Suppose that $\nu=\frac{1}{2}\delta_a+\frac{1}{2}\delta_b$, where $0<a\leq b<+\infty$. Let $E$ be the set of edges in $\mathbb{Z}^2$, and define $\mu=\nu^E$. For a resistance configuration sampled according to $\mu$, write $\mathcal{R}_r(0\leftrightarrow v)$ for the effective resistance between $0$ and $v$.

Bounded-variance conjecture. As $v$ tends to infinity,

$$\operatorname{Var}_\mu(\mathcal{R}_r(0\leftrightarrow v))=\Theta(1).$$

The conjecture asserts that the variance remains of constant order as the target vertex recedes to infinity. The source states that this is expected but not proved.