Vershik's uniqueness conjecture for the Poisson-Dirichlet distribution under CCF

Let $\Omega_1$ be the space of decreasing nonnegative sequences with sum $1$, and let $\widehat{\mu}_1$ denote the Poisson-Dirichlet distribution on $\Omega_1$ with parameter $\theta=1$. Let the CCF process be the continuous coagulation-fragmentation process arising as the scaling limit of the discrete coagulation-fragmentation chains under random transpositions. Vershik's conjecture. $\widehat{\mu}_1$ is the unique invariant distribution for the CCF. The Poisson-Dirichlet distribution with parameter $1$ is already known to be invariant; the conjecture concerns uniqueness of the invariant distribution for the limiting coagulation-fragmentation process.