Finite Andrews–Gordon companion hierarchy

Prove the polynomial identity in Conjecture 1 for all integers n≥0, k≥5, and 1≤i<k, relating the multiple q-binomial sum with doubled αij contributions to the stated alternating bilateral q-binomial sum.

Background

The paper establishes the identity for k=2, 3, and 4 and conjectures the displayed extension beginning at k=5. The conjectured formula is presented as a companion to the Foda–Quano polynomial refinement of the Andrews–Gordon identities and is motivated by computations with cylindric partitions of small profiles.

The conjecture concerns a finite polynomial refinement: its left-hand side is a multiple sum over n1≥⋯≥nk−1≥nk=0 with modified q-binomial factors, while its right-hand side is an alternating bilateral sum indexed by r. Establishing it would extend the verified low-k cases to the full infinite hierarchy.