The fish-scale conjecture for posets

Let $P$ be a poset containing no infinite antichain. A chain is a pairwise comparable subset of $P$, and an antichain is a pairwise incomparable subset. Fish-scale conjecture. There exist a chain $C$ and a decomposition of the vertex set into antichains $A_i$ such that $C$ meets every antichain $A_i$. The text identifies this as an open problem and notes that its dual follows from the infinite version of König’s theorem.