Fu–Peng–Zhang's coloring conjecture for finitary upho posets

An upho poset is a poset whose every principal filter is order-isomorphic to the whole poset; it is finitary when its intervals are finite. A coloring assigns labels to the covering edges of an upho poset in a way compatible with the principal-filter isomorphisms.

Fu–Peng–Zhang's coloring conjecture. Every finitary upho poset, in particular every finite-type $\mathbb{N}$-graded upho poset, has a coloring.

By the equivalence established in the paper, this is equivalent to the corresponding multiplicability statement. The conjecture is disproved by the paper's Petersen-graph construction, which yields a finitary finite-type $\mathbb{N}$-graded upho poset with no coloring.