Marić's selection principle for the Fleming–Viot random walk

Let $p$ and $q$ be the right- and left-jump rates, respectively, with $q>p$, and let $\rho^N$ be the equilibrium density of the Fleming–Viot process driven by the nearest-neighbor random walk on ${\mathbb N}$, absorbed at the origin. Let $\nu_0$ denote the minimal quasi-stationary distribution of the random walk. Marić's selection principle. As $N$ tends to infinity, the equilibrium density converges to the minimal quasi-stationary distribution:

$$\rho^N \to \nu_0.$$

This conjecture asserts that the Fleming–Viot system selects the minimal quasi-stationary distribution among the infinitely many quasi-stationary distributions of the random walk. Ergodicity of the Fleming–Viot process for $q>p$ is known, while the stated large-particle-number convergence is presented in the source as a conjecture supported by heuristic arguments and simulations.