Kirillov’s conjecture: Nonnegativity of c_{uv}^w(y;0)

Establish that for all permutations u, v, w in S_infty, the coefficients c_{uv}^w(y;0), defined by sch_u(x;y) sch_v(x;0) = sum_w c_{uv}^w(y;0) sch_w(x;y), are polynomials in the variables y with nonnegative integer coefficients.

Background

Kirillov proposed (essentially) that the specialization z=0 of the Molev–Sagan coefficients is nonnegative as a polynomial in y. This conjecture is a special case of the broader positivity conjecture c_{uv}^w(y;z) ≥ 0 in the sense of polynomials in y_i − z_j. Kirillov proved the conjecture when the Bruhat length difference ℓ(u,w)=1. The authors report extensive computational verification (for u,v in S_7) but no general proof is known.