Strong sign-coherency of certain symmetric polynomials, with application to cluster algebras

For each positive integer n, we define a polynomial in the variables z_1,…,z_n with coefficients in the ring $\mathbb{Q}[q,t,r]$ of polynomial functions of three parameters q, t, r. These polynomials naturally arise in the context of cluster algebras. We conjecture that they are symmetric polynomials in z_1,…,z_n, and that their expansions in terms of monomial, Schur, complete homogeneous, elementary and power sum symmetric polynomials are sign-coherent.