Skyline-tableau formula for Lascoux atoms

Let $\beta$ be a parameter, let $\boldsymbol{z}=(z_1,z_2,\ldots)$, let $\lambda$ be a partition, and let $w$ be a permutation. Let $\operatorname{SSLT}(w\lambda)$ denote the set of semistandard skyline tableaux of content or weight $w\lambda$, with weight $\operatorname{wt}(S)$ and excess $\operatorname{ex}(S)$. Let $\overline{L}_{w\lambda}$ denote the Lascoux atom.

Skyline-tableau conjecture. We have

$$\overline{L}_{w\lambda} = \sum_{S \in \operatorname{SSLT}(w\lambda)} \beta^{\operatorname{ex}(S)} \boldsymbol{z}^{\operatorname{wt}(S)}.$$

This is presented as the K-theoretic analogue of the description of Demazure characters by skyline tableaux. The source later proves the conjecture, so the tableau expansion is established.