Finite Bressoud companion hierarchy

Prove the polynomial identity in Conjecture 2 for all nonnegative integers n and all integers k≥5 with 1≤i<k, relating the modified multiple q-binomial sum to the stated alternating bilateral q-binomial sum.

Background

The paper proves the corresponding identity for k=2, 3, and 4 and proposes the displayed formula for k≥5. It is designed as a companion to the Foda–Quano polynomial refinement of Bressoud identities.

The conjectured identity modifies both the terminal q^2-binomial factor and the other q-binomial factors through the doubled αij contribution. A proof would extend the finite Bressoud hierarchy beyond the cases accessible by the paper’s direct MacMahon-analysis and computer-algebra arguments.