The $K$-Kohnert rule for dual stable Grothendieck polynomials

Let $\alpha$ be a weak composition, let $J_{\alpha}^{(\beta)}$ be the generating function for diagrams produced from $\operatorname{Skyline}(\alpha)$ by the $K$-Kohnert rule, with each diagram weighted by $\beta^{g(D)}x^D$, where $g(D)$ is the number of $g$'s in $D$ and $x^D$ is its column weight. Let $\Omega_{\alpha}$ denote the dual stable Grothendieck polynomial associated with $\alpha$. The $K$-Kohnert conjecture.

$$J_{\alpha}^{(-1)}=\Omega_{\alpha}.$$

This conjecture extends Kohnert's rule from key polynomials to the dual stable Grothendieck basis by incorporating marked, unmovable $g$'s. The source provides no resolution or evidence of a proof, so its status remains open.