The asymptotic shape conjecture for effective-resistance balls in the square lattice

Let $r$ be an i.i.d. resistance configuration on the edges of $\mathbb{Z}^2$, and write $\mathcal{R}_r(0\leftrightarrow v)$ for the effective resistance between $0$ and $v$. Define

$$B_t=\\{v\in\mathbb{Z}^2: \mathcal{R}_r(0\leftrightarrow v)\leq t\\}.$$

Asymptotic shape conjecture. There exists a nonempty compact subset $B_0$ of $\mathbb{R}^2$ such that, for every positive number $\varepsilon$,

$$(1-\varepsilon)B_0\subseteq \frac{1}{\log t}B_t\subseteq (1+\varepsilon)B_0.$$

The claim predicts an asymptotic shape for effective-resistance balls, analogous to the time constant and limit shape in first-passage percolation. The source presents the existence of the time constant and asymptotic shape as believed but does not provide a resolution.