Addario-Berry et al.'s critical resistance scaling conjecture

Let $R_n$ denote the resistance in the random resistance model, let $p_c=\frac12$, and let $y\in\mathbb R$. Addario-Berry et al.'s conjecture. If $p=p_c=\frac12$, there exists a constant $c_{\mathrm{eff}}\in(0,\infty)$ such that

$$\lim_{n\to\infty}\mathbb P\left(\frac{\log R_n}{(c_{\mathrm{eff}}n)^{1/3}}\le y\right)=\int_{-\infty}^y\frac34(1-x^2)\mathbf 1_{\{|x|<1\}}\,\mathrm dx.$$

The prediction gives a precise $n^{1/3}$ critical scaling and would imply the Hambly–Jordan conjecture. The source presents it as an expectation rather than a proved result.