Complete convergence conjecture for voter model perturbations

Let $\alpha=(\alpha_0,\alpha_1)$ be in the coexistence region of Theorem 1.10 of CDP11, with each $\alpha_i<1$ and sufficiently close to $1$. Let $\nu_{1/2}$ denote the nontrivial stationary distribution appearing in the complete convergence theorem, and let $\nu_\alpha$ be a stationary distribution for the corresponding voter model perturbation. Complete convergence conjecture. For such $\alpha$, the complete convergence theorem holds with a unique nontrivial stationary distribution $\nu_\alpha$ in place of $\nu_{1/2}$. This conjecture predicts that the complete convergence behavior persists throughout the coexistence region for perturbations sufficiently close to the voter model; the supplied text does not state that it has been resolved.