The Aharoni–Korman conjecture for ordered sets without infinite antichains

Let $P$ be an ordered set with no infinite antichains, and let $k$ be a positive integer. Aharoni and Korman's conjecture asserts that there are $k$ chains $C_1,\ldots,C_k$ and a partition of $P$ into antichains $(A_i:i\in I)$ such that each $A_i$ intersects $\min(|A_i|,k)$ of the chains $C_j$. The conjecture generalizes the assertion that one chain can meet every part of a partition into antichains. The instance $k=1$ is known for well-founded ordered sets with every level finite; the paper verifies the full conjecture for posets whose incomparability graph is locally finite, and consequently for $(3+1)$-free posets with no infinite antichains.