Eventual positivity and unimodality

Show that for every balanced coprime pair of positive-integer tuples satisfying Landau’s criterion, there exists a positive integer N such that the associated q-factorial ratios D_n(a,b;q) are positive and unimodal for all n greater than or equal to N.

Background

The paper concludes by posing an asymptotic version of the positivity and unimodality questions. Extensive computations suggest that D_n(a,b;q) is unimodal for sufficiently large n in many examples, motivating a proof of eventual behavior rather than an immediate proof for every n.

The problem has two parts: first, establish the existence of a threshold N for each balanced coprime pair satisfying Landau’s criterion; second, obtain an explicit upper bound for that threshold in terms of the pair.