Nonnegativity conjecture for double Schubert structure constants

Let $u,v,w$ be permutations, and define the coefficients $c_{uv}^w(y;z)$ by

$$\operatorname{\mathfrak{S}}_u(x;y)\operatorname{\mathfrak{S}}_v(x;z)=\sum_{w\in S_\infty}c_{uv}^w(y;z)\operatorname{\mathfrak{S}}_w(x;y).$$

Nonnegativity conjecture. For all $u,v,w$, $c_{uv}^w(y;z)$ is a polynomial in the differences $y_i-z_j$ with nonnegative integer coefficients.

The conjecture generalizes positivity results for factorial Schur functions and specializes at $y=z$ to known equivariant positivity statements. It has been computationally verified for $u,v\in S_5$, but remains open in general; related specializations and geometric cases are known.