Aharoni–Korman fish bone conjecture for bounded-width posets

Let $P$ be a poset of bounded width: there is a finite number $k$ such that every antichain $A$ satisfies $|A|\leq k$. A chain is a totally ordered subset of $P$, and a decomposition of $V(P)$ into disjoint antichains is a partition of the elements of $P$ into antichains.

Fish bone conjecture. There exist a chain $C$ and a decomposition $D$ of $V(P)$ into disjoint antichains such that every antichain in $D$ meets $C$.

This conjecture is trivial for finite posets and follows from the edge-cover version of the preceding hypergraph conjecture together with compactness. It is known for $k=2$, but remains open in general; the source notes that even the case with no infinite antichains has no known counterexample.