The recurrence conjecture for the positive-voter mixture

Let $\xi$ be the exclusion-voter mixture process with parameters $\beta,p$ and state space $\mathcal{D}$. A process is recurrent from $S_0$ when ${\mathbb{P}}_{\beta,p}(\tau<\infty\mid\xi_0=S_0)=1$, where $\tau$ is the relaxation time to the distinguished state $\mathcal{D}_0$. The recurrence conjecture. Suppose that $p<1/2$ and $\beta>0$. For any $S_0\in\mathcal{D}$, $\xi$ is recurrent. The paper presents this as an open question contrasting recurrence already known for $p\geq1/2$ with the unresolved behavior for $p<1/2$ and $\beta>0$; simulations were inconclusive.