Converse characterisation of separately exchangeable random matrices by empirical distributions

Let $X=(X_{i,j})$ be a finite or infinite random matrix taking values in a Borel space $S$, and define its empirical distributions by

$$\eta_{n,m}=\frac{\sum_{j\leq m}\sum_{i\leq n}\delta_{X_{i,j}}}{nm}.$$

A collection of random measures $(\eta_{n,m})_{(n,m)\in\mathbb{N}^2}$ is a reverse measure-valued martingale if, for every $n\geq k$, $m\geq l$, and measurable function $f$,

$$\eta_{n,m}f=E(\eta_{k,l}f\mid\theta_{n-1,m-1}\eta).$$

Converse characterisation conjecture. If the empirical distributions $(\eta_{n,m})$ form a reverse measure-valued martingale, then $X$ is separately exchangeable, meaning that for any permutations $p_1,p_2$ of $\mathbb{N}$,

$$X\circ(p_1,p_2)\stackrel{d}{=}X.$$

This is the converse to the reverse-martingale property established for separately exchangeable matrices. The supplied text does not indicate whether the converse has been proved or remains open.