Positivity of c_{uv}^w(y;z) in the Molev–Sagan product

Prove that for all permutations u, v, w in the infinite symmetric group S_infty, the structure coefficient c_{uv}^w(y;z), defined by the expansion of the product of double Schubert polynomials sch_u(x;y) sch_v(x;z) = sum_w c_{uv}^w(y;z) sch_w(x;y), is a polynomial in the differences y_i − z_j with nonnegative integer coefficients.

Background

Double Schubert polynomials sch_u(x;y) form a basis over Z[y] and encode equivariant cohomology classes of the complete flag variety. Introducing a second set of coefficient variables z, the product sch_u(x;y) sch_v(x;z) expands in the sch_w(x;y) basis with coefficients c_{uv}^w(y;z).

The conjecture asserts a strong positivity property for these coefficients as polynomials in the differences y_i − z_j. This generalizes known positivity phenomena: c_{uv}^w(0;0) ≥ 0 holds by geometric arguments in ordinary cohomology; c_{uv}^w(y;y) is Graham-positive (polynomial in simple negative roots with nonnegative coefficients) by Graham (2001); and the factorial Schur case is positive via the Molev–Sagan Littlewood–Richardson rule. The paper also verifies the conjecture computationally for small permutations.