q-rious unimodality conjecture

Prove that whenever a pair of positive-integer tuples (a, b) satisfies Landau’s criterion, the polynomial (1 + q)D(a, b; q) is unimodal.

Background

The q-factorial ratio D(a, b; q) has a symmetric coefficient sequence under Landau’s criterion. The proposed q-rious unimodality conjecture asserts unimodality after multiplication by 1 + q, a property that would imply nonnegativity of the coefficients of D(a, b; q) and hence the q-rious positivity conjecture.

The conjecture is verified for Bober’s 52 sporadic irreducible coprime pairs and one two-parameter family, while its validity remains unproved in several important parametric families.