Unimodality for the remaining two-parameter families

Prove that the q-factorial ratios associated with the two-parameter families (5b) and (5c) are unimodal, with the factor 1 + q as specified by Conjecture 5.

Background

The two families in question are the q-factorial ratios corresponding to the tuples (5b) and (5c). They are among the principal nontrivial parametric examples discussed after the established q-binomial family.

The authors report that extensive computations support the general q-rious unimodality conjecture, but they have not been able to prove it for these two families. For the q-super Catalan family (6c), they further suggest that unimodality may hold without the factor 1 + q, whereas the factor appears necessary in some exceptional cases of (6b).

References

This, combined with the lack of a combinatorial or representation theoretic interpretation of (6b) and (6c), perhaps explains why so far we have been unable to show that the conjecture holds for the pairs (5b) and (5c).

— $q$-rious unimodality  (2502.03993 - Warnaar et al., 6 Feb 2025) in Remark following Lemma 7, Introduction