The Borel dual Dilworth conjecture

Let $P=(V,\preceq)$ be a Borel poset. Its height is the largest size of a chain in $P$, and an antichain is a set of pairwise incomparable elements. Borel dual Dilworth conjecture. Every Borel poset $P$ of finite height $h$ is decomposable into $h$ Borel antichains. This is the dual Borel analogue of Dilworth's theorem. The paper presents the statement as a conjectural extension; its general validity is not established there.