Non-negativity conjecture for orbit-counting polynomials of pairs in abelian groups

Let $R$ be a discrete valuation ring with residue field of order $q$, let $M$ be the finite abelian group associated with a partition $\lambda$, and let $G$ act on $M\times M$ as in the paper. Write $n_\lambda(q)$ for the polynomial representing the number of $G$-orbits in $M\times M$. Non-negativity conjecture. The polynomial $n_\lambda(q)$ has non-negative coefficients. This conjecture predicts a stronger positivity property than the established fact that $n_\lambda(q)$ is monic with integer coefficients of degree $\lambda_1$; the paper reports computational evidence but does not provide a proof.