Subpolynomial maximal resistance in degree-two fractal mother graphs

Let $\mathcal G(2,m,n)$ be the finite fractal mother graph with branching parameter $m$ and level $n$, and let its maximal resistance mean the maximum effective resistance between the relevant vertices of the graph.

Subpolynomial resistance conjecture. The maximal resistance in $\mathcal G(2,m,n)$ grows slower than any power of $n$:

$$\text{for every }\alpha>0,\quad R_{\max}(\mathcal G(2,m,n))=o(n^\alpha).$$

The authors contrast this with the degree-one case and state that they will prove recurrence of $\mathcal G(2,m,\infty)$ in future work; the subpolynomial resistance assertion itself remains unproved here.