The measurable Dilworth conjecture

Let $P=(V,\preceq)$ be a measurable poset, meaning an ordered probability space with probabilistic measure $\mu$. Let $G_P$ be its incomparability graph, and let $\chi_{\mu}(G_P)$ denote the measurable chromatic number of $G_P$. Measurable Dilworth conjecture. Every measurable poset $P$ of finite width $k$ is decomposable, up to a null set, into $k$ measurable chains. Equivalently,

$$\chi_{\mu}(G_P)=k.$$

This is a measurable analogue of the Borel Dilworth problem. The conjecture is motivated by the result that atomless measurable posets can be included in realistic measurable posets, but the general assertion remains open.