Kohnert-diagram formula conjecture for Lascoux polynomials

Let $a=(a_1,otsc,a_n)in\boldsymbol{Z}_{\geq0}^n$. A K-Kohnert diagram is a subset $D$ of $\boldsymbol{Z}_{>0}^n$ together with a subset of marked boxes, obtained from the skyline diagram for $a$ by Kohnert moves and K-Kohnert moves. Define its $\beta$-weight by

$$\operatorname{wt}_{\beta}(D)=\beta^e\prod_{i=1}^n x_i^{c_i},$$

where $e$ is the number of marked boxes and $c_i$ is the number of boxes in column $i$. Kohnert-diagram formula conjecture. The Lascoux polynomial satisfies

$$L_a(\mathbf{x};\beta)=\sum_{D\in\mathcal{D}_a}\operatorname{wt}_{\beta}(D).$$

This is a conjectural combinatorial rule for Lascoux polynomials, extending Kohnert's rule at $\beta=0$ and incorporating marked boxes to capture the K-theoretic deformation.