  In previous work, we constructed Fleming--Viot-type measure-valued diffusions
(and diffusions on a space of interval partitions of the unit interval $[0,1]$)
that are stationary with the Poisson--Dirichlet laws with parameters
$\alpha\in(0,1)$ and $\theta\geq 0$. In this paper, we complete the proof that
these processes resolve a conjecture by Feng and Sun (2010) by showing that the
processes of ranked atom sizes (or of ranked interval lengths) of these
diffusions are members of a two-parameter family of diffusions introduced by
Petrov (2009), extending a model by Ethier and Kurtz (1981) in the case
$\alpha=0$. The latter diffusions are continuum limits of up-down Chinese
restaurant processes.
