Eventual positivity or unimodality of factorial-ratio families

Establish that for every balanced coprime pair of positive-integer tuples satisfying Landau's criterion, there exists a positive integer N such that D_n(a,b;q) is positive or unimodal for every n≥N, and determine an explicit upper bound for N in terms of (a,b).

Background

The authors observe that extensive computational checks suggest that D_n(a,b;q) is unimodal for all integers n>2 across many sporadic and parametric families, including examples arising from root systems and basic hypergeometric summation formulas.

Because proving unimodality, and in some cases even positivity, is difficult, the paper concludes with a weaker asymptotic problem. It asks first for eventual positivity or unimodality for every balanced coprime Landau pair and second for an explicit bound on the threshold beyond which the property holds.