Monomial-count conjecture for symmetrized Mordell–Tornheim zeta values

Let $\overline{\zeta}_n$ be the symmetrized Mordell–Tornheim zeta value of order $n$, and let $p(n)$ denote the partition function. A monomial has total weight $n$ when the sum of the weights of its zeta-function factors is $n$. Monomial-count conjecture. The value of $\overline{\zeta}_{n\geq 1}$ is a homogeneous polynomial with $p(n)-p(n-1)$ monomials, where each monomial is a product of zeta function values with total weight equal to $n$. This predicts that all such monomials occur with nonzero coefficients; its validity for arbitrary $n$ remains open.