Conditional expectation conjecture for contractive projections on rearrangement-invariant spaces

Let $X$ be a rearrangement-invariant function space on $[0,1]$, so that $X$ contains the constant functions, and let $P$ be a contractive projection on $X$ satisfying $P(\boldsymbol{1})=\boldsymbol{1}$. Conditional expectation conjecture. There exists a $\sigma$-algebra $\Sigma_0$ of measurable subsets of $[0,1]$ such that $P$ is the conditional expectation operator

$$\mathcal{E}^{\Sigma_0}.$$

This would generalize the corresponding characterization for $L_p$ spaces to rearrangement-invariant function spaces. The source states that the converse of the conditional-expectation theorem is not known in this generality.