The Borel Dilworth conjecture

Let $P=(V,\preceq)$ be a Borel poset, meaning that $V$ is a standard Borel space and the comparability relation is a Borel subset of $V\times V$. Its width is the maximum size of an antichain. Borel Dilworth conjecture. Every Borel poset $P$ of finite width $k$ is decomposable into $k$ Borel chains. This is the Borel analogue of Dilworth's theorem for finite posets; the paper proves the assertion only in special cases, so the general statement remains open.