Law of large numbers for the range of rotor walks in the null recurrent case

Let $\mathsf{G}$ be a finite graph on $\mathsf{N}$ vertices, let $D$ be its adjacency matrix, and let $\mathsf{T}_i$ be a periodic tree directed cover of $\mathsf{G}$ with root of type $i\in\{1,\ldots,\mathsf{N}\}$. Let $(X_n^i)_{n\geq 0}$ be a rotor walk on $\mathsf{T}_i$ with a $\mathcal{D}$-distributed random initial rotor configuration, let $R_n^i=\{X_1^i,\ldots,X_n^i\}$ be its range, and let $M$ be the first moment matrix of the multitype branching process formed by the tree of good children. Here $\rho(M)$ denotes the spectral radius of $M$. Null-recurrent range conjecture. For all $i\in\{1,\ldots,\mathsf{N}\}$, if $\rho(M)=1$, then

$$\frac{|R_n^i|}{n}\to \frac{1}{2},\quad\text{almost surely, as } n\to\infty.$$

This conjecture predicts the asymptotic range growth of rotor walks in the null recurrent case, complementing the proved law of large numbers for the transient case $\rho(M)<1$.