Unimodality for the remaining classified two-parameter families

Show that the q-factorial ratios associated with the two-parameter families (5b) and (5c), namely the q-analogue of the super Catalan polynomials and the third family listed in equations (6b) and (6c), satisfy Conjecture 5.

Background

Lemma 7 establishes Conjecture 5 for Bober’s 52 sporadic irreducible coprime pairs and for family (5a), the q-binomial family. The paper then discusses the two remaining two-parameter families (5b) and (5c), whose coefficient unimodality is not established.

The authors note that these families lack the combinatorial or representation-theoretic interpretations available for q-binomial coefficients, which contributes to the difficulty of proving unimodality. For the q-super Catalan family, they further suggest that the factor 1+q may not be necessary in most cases, but this stronger claim is not proved.