Small-voter-fraction nonergodicity conjecture for the hybrid process

Let $p$ be the exclusion-process parameter, let $\beta$ be the voter proportion, and let $\tilde\eta_t$ denote the hybrid process with parameters $(\beta,p)$. A process is called ergodic when it has the ergodicity property considered in the paper.

Small-voter-fraction nonergodicity conjecture. For every $p<1/2$, there exists a number $\beta_0(p)>0$ such that, whenever $\beta<\beta_0(p)$, the hybrid process $\tilde\eta_t$ with parameters $\beta$ and $p$ is not ergodic.

The conjecture asserts that a sufficiently small voter component cannot prevent the exclusion component from escaping when its drift parameter satisfies $p<1/2$. The paper provides ergodicity results for sufficiently large voter proportions and for the symmetric exclusion case, but leaves these sufficient conditions for nonergodicity open.