Reversibility conjecture for Fleming–Viot diffusions

Let $\tt FV(\alpha,\theta)$ be the Fleming–Viot diffusion with parameters $\alpha\in(0,1)$ and $\theta>-\alpha$, and let $\tt PDRM(\alpha,\theta)$ denote its stationary Poisson–Dirichlet random measure distribution. Reversibility conjecture. The process $\tt FV(\alpha,\theta)$ is reversible with respect to $\tt PDRM(\alpha,\theta)$. The ranked sequence of atom sizes is known to be reversible because of the reversibility of the corresponding EKP diffusion; the conjecture asks whether the full labeled Fleming–Viot diffusion has the same reversibility property.