Nonnegativity of the conjectured bilateral sums

Prove that, for every k≥5, k≥i≥1, and n≥0, both alternating bilateral q-binomial sums displayed in the conjecture have non-negative q-series coefficients.

Background

The finite bilateral expressions obtained from cylindric-partition generating functions are often written as alternating sums, so coefficientwise nonnegativity is not manifest from their formulas. The paper proves positivity in several low-parameter cases by identifying alternative manifestly positive multiple-sum representations.

The conjecture asks for coefficientwise nonnegativity of the generalized Andrews–Gordon- and Bressoud-type sums in the unproved range k≥5. Such a result would clarify the combinatorial positivity underlying the alternating expressions.