Fleming–Viot invariant-measure conjecture for Yaglom limits

Let $Z=(Z_t,t\geq0)$ be a Markov driving process on $\Lambda\cup\{0\}$, with $0$ absorbing, and let $\tau$ be its absorption time. For $N\geq2$, let $\xi_t=(\xi_t(1),\ldots,\xi_t(N))\in\Lambda^N$ be the Fleming–Viot process driven by $Z$, and define its empirical measure by

$$\mu_t^N=\frac1N\sum_{i=1}^N\delta_{\xi_t(i)}.$$

A Yaglom limit is a probability measure $\nu$ describing the limiting law of $Z_t$ conditioned on survival. Fleming–Viot invariant-measure conjecture. If the driving process $Z$ has a Yaglom limit $\nu$, then the Fleming–Viot process driven by $Z$ is ergodic, with a unique invariant measure $\lambda^N$, and the empirical measures distributed according to $\lambda^N$ converge to $\nu$. The finite-time convergence of the empirical measures to the conditioned evolution is stated as proved in the source, while this fixed-$N$ ergodicity and invariant-measure limit is presented as an expected general principle.