Recurrence conjecture for IID $p$-rotor walks

Let $p\in[0,1]$. On $\mathbb{Z}^2$, a $p$-rotor walk is the walk in which, at each step, the rotor at the walker's current location is rotated $90$ degrees counterclockwise with probability $p$ and $90$ degrees clockwise with probability $1-p$. Let $\rho$ be sampled from the IID uniform measure on $\{\operatorname{\mathcal{H}},\operatorname{\mathcal{V}}\}$.

$p$-rotor recurrence conjecture. For almost every such $\rho$, the corresponding $p$-rotor walk on $\mathbb{Z}^2$ visits every vertex infinitely many times almost surely.

The claim extends the recurrence question across the full parameter range $p\in[0,1]$, including the horizontal-vertical walk at $p=\tfrac12$. The supplied text gives no resolution and explains that the methods in the paper are sensitive to changes in the model.