Kohnert's conjecture for Grothendieck polynomials

Let $w\in S_n$, let $\operatorname{Rothe}(w)$ be the Rothe diagram of $w$, and generate diagrams $D$ from it using the $K$-Kohnert rule, in which each $+$ either moves according to Kohnert's rule or stays in place and moves while its original position is marked by an unmovable $g$. Define

$$K_w^{(\beta)}=\sum_D \beta^{g(D)}\mathbf{x}^D,$$

where $g(D)$ is the number of $g$'s in $D$. Let $\mathfrak G_w$ be the Grothendieck polynomial indexed by $w$. Kohnert's conjecture.

$$K_w^{(-1)}=\mathfrak G_w.$$

This is the $K$-theoretic extension of Kohnert's rule, whose unmarked version generates Schubert polynomials from Rothe diagrams. The source gives no resolution or status evidence for the conjecture, so it remains open.