The Aharoni--Korman conjecture for posets whose incomparability graph is locally finite

Aharoni and Korman (Order 9 (1992) 245–253) have conjectured that every ordered set without infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain. The conjecture is verified for posets whose incomparability graph is locally finite. It follows that the conjecture is true for $(3+1)$-free posets with no infinite antichains.