Spines in finite-width posets

Determine whether every poset P of finite width w admits a spine; that is, determine whether there exists a chain C in P and a partition of P into antichains (A_i) such that C meets every antichain A_i in the partition.

Background

A spine is a central object in the Aharoni–Korman framework: a chain intersecting every antichain in a partition of the poset. The paper constructs a counterexample to the Aharoni–Korman conjecture in full generality, but that counterexample has infinite width, which motivates investigating whether finite-width posets always admit spines.

It is known that posets of width 2 have spines, but beyond width 2 the problem remains unresolved; even width 3 is open in the countable case.