The transience conjecture for uniform shifts on regular trees

Let $\mathbb{T}_{d}$ be the homogeneous $d$-ary tree with $d\geq 3$, and let ${\mathbf a}$ be an $L$-periodic rotor sequence. The regular-tree uniform-shift transience conjecture. The rotor walk in the corresponding uniform shift model is transient almost surely for every ${\mathbf a}$. The preceding lemma proves this assertion for the relevant class of sequences, but the stated conjecture for arbitrary periodic rotor sequences remains open.