Strict positivity of coefficients of the partition-indexed polynomials

Prove that, for every integer partition λ, all coefficients of the polynomial J_λ(q) are strictly positive.

Background

The paper studies polynomials J_λ(q) indexed by integer partitions λ. These polynomials generalize the inversion enumerator polynomials J_n and are known to have integer coefficients, but a general combinatorial interpretation is not established. The authors report that strict positivity has been verified in some cases, including all partitions of size at most 43, and that the corresponding polynomials J_n^r satisfy analogous positivity properties. They identify a combinatorial interpretation of J_λ as an attractive possible route toward proving the conjecture.