The positivity conjecture for Grothendieck polynomials in type $A$

Let $S_n$ be the symmetric group, let ${\cal G}_w(x_1,\ldots,x_n)$ be the Grothendieck polynomial associated with $w\in S_n$, and let $G_i^{A_{n-1}}$ be the elements defined by $G_i^{A_{n-1}}=\Theta_i^{A_{n-1}}-1$, where the $\Theta_i^{A_{n-1}}$ are the commuting elements of $BE(A_{n-1})$. The positivity conjecture for Grothendieck polynomials in type $A$. For every permutation $w\in S_n$, after substituting $x_i:=G_i^{A_{n-1}}$ for $1\leq i\leq n$ and setting $z=1$, the value of ${\cal G}_w(x_1,\ldots,x_n)$ can be written as a linear combination of monomials in the $x_{ij}$, $1\leq i<j\leq n$, with non-negative integer coefficients. This is a positivity assertion connecting Grothendieck-polynomial calculus with the bracket algebra; the source gives no resolution of the conjecture.