Graham positivity conjecture for triple Grothendieck polynomial coefficients

Let $\beta$ be a formal variable, let $u,v,w\in S_\infty$, and let $\mathfrak{G}_w(\mathbf{x};\mathbf{y})$ denote the double Grothendieck polynomial. Define the coefficients $\tilde{c}_{u,v}^{\,w}(\mathbf{y},\mathbf{t})$ by

$$\mathfrak{G}_u(\mathbf{x};\mathbf{y})\mathfrak{G}_v(\mathbf{x};\mathbf{t})=\sum_{w\in S_\infty}\tilde{c}_{u,v}^{\,w}(\mathbf{y},\mathbf{t})\mathfrak{G}_w(\mathbf{x};\mathbf{t}).$$

For variables $a,b$, write $a\ominus b:=\frac{a-b}{1+\beta b}$. Graham positivity conjecture. For all $u,v,w\in S_\infty$,

$$\tilde{c}_{u,v}^{\,w}(\mathbf{y},\mathbf{t})\in\mathbb{N}[\beta][t_i\ominus y_j]_{i,j\geq 1}.$$

This conjecture proposes a positivity property for the coefficients in the product of double Grothendieck polynomials, extending the Graham-positivity phenomenon established earlier in the paper for triple Schubert calculus. Its resolution is not supplied in the given text.