Eventual positivity and unimodality of q-factorial ratios

Show that for every balanced coprime pair of positive-integer tuples (a, b) satisfying Landau’s criterion, there exists a positive integer N such that D_n(a, b; q) is positive and unimodal for every n ≥ N.

Background

The paper concludes with an asymptotic problem concerning the dilated q-factorial ratios D_n(a, b; q) obtained by replacing every tuple entry with its multiple by n. The proposed result would establish positivity and unimodality not necessarily for every n, but uniformly for all sufficiently large n.

This problem is presented as potentially more accessible through asymptotic methods than the full q-rious unimodality conjecture. It is restricted to balanced coprime pairs satisfying Landau’s criterion.

References

Problem. Let (a, b) be any balanced coprime pair satisfying Landau's criterion (1). 1. Show that there exists a positive integer N such that D_n (a, b; q) is positive/unimodal for all n ≥ N. 2. Find an explicit upper bound for N in terms of (a, b).

— $q$-rious unimodality  (2502.03993 - Warnaar et al., 6 Feb 2025) in Final section, concluding Problem

Problem. Let (a, b) be any balanced coprime pair satisfying Landau's criterion (1). 1. Show that there exists a positive integer N such that Dn (a, b; q) is positive/unimodal for all n ≥ N.

— $q$-rious unimodality  (2502.03993 - Warnaar et al., 6 Feb 2025) in Problem, Conclusion