Set-valued tableau formulas for Lascoux polynomials and 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. Write $\operatorname{SVT}^n(\lambda)$ for the set of set-valued tableaux of shape $\lambda$ with entries bounded by $n$, $\operatorname{wt}(T)$ for the weight of $T$, and $\operatorname{ex}(T)$ for its excess. Let $K_{w\lambda}$ denote the key tableau associated to $w\lambda$, and compare tableaux entrywise. The Lascoux polynomial and atom are denoted by $L_{w\lambda}(\boldsymbol{z};\beta)$ and $\overline{L}_{w\lambda}(\boldsymbol{z};\beta)$, respectively.

Set-valued tableau conjecture. We have

$$\overline{L}_{w\lambda}(\boldsymbol{z}; \beta) = \sum_{\substack{T \in \operatorname{SVT}^n(\lambda) \\ K(T) = K_{w\lambda}}} \beta^{\operatorname{ex}(T)} \boldsymbol{z}^{\operatorname{wt}(T)}, \qquad\qquad L_{w\lambda}(\boldsymbol{z}; \beta) = \sum_{\substack{T \in \operatorname{SVT}^n(\lambda) \\ K(T) \leq K_{w\lambda}}} \beta^{\operatorname{ex}(T)} \boldsymbol{z}^{\operatorname{wt}(T)}.$$

At $\beta=0$, the analogous formulas were known from Lascoux and Schützenberger. The conjecture gives a positive tableau expansion for general Lascoux polynomials and atoms; the source later proves it.