Finite copartition-product non-negativity conjecture

Prove that for all positive integers a, b, m, N, and M satisfying b divides a, a+b=m, and N\leq M, the finite copartition product g_{a,b,m}(N,M;q)=(-q^m;q^m)_{N+M-1}/[(-q^a;q^m)_N(q^b;q^m)_M] has only non-negative coefficients, equivalently g_{a,b,m}(N,M;q)\in\mathbb{N}_0[[q]].

Background

The finite product g_{a,b,m}(N,M;q) was introduced as a finite analogue of the infinite copartition product. The conjecture asserts coefficientwise non-negativity under the divisibility, balancing, and truncation conditions b\mid a, a+b=m, and N\leq M.

The paper reduces this conjecture to the diagonal case and proves it when N=1 or a=b. It also proves non-negativity through an explicit initial degree range and eventual positivity properties, but does not settle the conjecture in general.