Chain and independent-set partition conjecture for posets of bounded width

Let $P$ be a poset of bounded width, viewed through its comparability structure. A chain is a subset of pairwise comparable elements, and an independent set is a subset containing no comparable pair. Chain and independent-set partition conjecture. There exists a chain $C$ in $P$ and a partition of the vertex set into independent sets, with every part of the partition meeting $C$. The source presents this as a conjecture that would follow from a positive answer to the strongly minimal cover conjecture; its status is therefore open.