Characterization of posets with coefficientwise nonnegative q-Zeta numerators

Characterize the finite graded posets with a unique maximum whose q-Zeta numerators are coefficientwise nonnegative under the paper’s fixed normalization, extending beyond the sufficient condition supplied by interval R-labellings.

Background

The paper establishes a sufficient positivity criterion: if the minimum augmentation of a finite graded poset with a unique maximum admits an R-labelling, then its q-Zeta numerator is coefficientwise nonnegative. The authors explicitly state that this criterion is not known to characterize all posets with positive q-Zeta numerators and ask for a broader classification.