The recurrence classification conjecture for uniform shifts on the binary tree

Let $\mathbb{T}_{2}$ be the binary tree, and let ${\mathbf a}$ be an $L$-periodic rotor sequence. Define $\mathcal{A}_{\mathrm{rec}}^{\mathrm{conj}}$ to consist of all shifts of sequences of the form

$$\overline{(0,i_{1},j_{1},0,i_{2},j_{2},\ldots,0,i_{N},j_{N})},$$

where $\{i_{\ell},j_{\ell}\}=\{1,2\}$ for $1\leq \ell\leq N$. The binary-tree uniform-shift recurrence conjecture. The rotor walk in the uniform shift model corresponding to ${\mathbf a}$ is recurrent almost surely if and only if ${\mathbf a}\in\mathcal{A}_{\mathrm{rec}}^{\mathrm{conj}}$. This conjecture was verified computationally for $L\leq 12$, but its general validity remains open.