Transience of the degree-three, two-parameter fractal mother graph

The graphs $\mathcal G(d,m,n)$ have infinite versions, and $\mathcal G(d,m,\infty)$ denotes the component containing the vertex $\cdots 000$. A graph is transient when the associated random walk is transient.

Transience conjecture. The graph $\mathcal G(3,2,\infty)$ is transient.

This would settle the missing case $d=3$, $m=2$; the paper proves transience for $d\geq 4$ or for $d\geq 3$ and $m\geq 3$.