Quenched Brownian scaling conjecture for IID horizontal-vertical rotor walks

Let $q>0$, and let $(X_t,\rho_t)_{t\geq 0}$ be an $\operatorname{\mathcal{H}}$--$\operatorname{\mathcal{V}}$ walk with a single walker, whose initial rotor configuration $\rho$ is sampled from the IID uniform measure on $\{\operatorname{\mathcal{H}},\operatorname{\mathcal{V}}\}$. Let $(B_t)_{t\geq 0}$ denote standard Brownian motion in $\mathbb{R}^2$.

Quenched scaling conjecture. For almost every such $\rho$,

$$\frac{1}{\sqrt{n}}(X_{\lfloor nt\rfloor})_{t\geq 0}\ \mathop{\Longrightarrow}_{n\to\infty}\ (B_t)_{t\geq 0},$$

with convergence weakly in the Skorohod space $D_{\mathbb{R}^2}[0,\infty)$.

The conjecture is motivated by simulations and by the known Brownian scaling limit for the stationary uniform-spanning-tree configuration. The supplied text notes that the case $q=0$ follows from work of Berger and Deuschel, while the case $q>0$ remains open.