Reiner–Yong conjecture on Grothendieck-to-Lascoux expansions

Let $w$ be an element of $S_+$, and let $\mathrm{Inc}_\lambda^w$ denote the set of increasing tableaux analogous to $\mathrm{Dec}_\lambda^w$. For an increasing tableau $P$, let $K_-(P)$ be its left key constructed using K-jeu-de-taquin, and let $\mathfrak{G}^{(\beta)}_w$ and $\mathfrak{L}^{(\beta)}_\alpha$ denote the Grothendieck and Lascoux polynomials, respectively.

Reiner–Yong conjecture.

$$\mathfrak{G}^{(\beta)}_w = \sum_\lambda \sum_{P\in \mathrm{Inc}_\lambda^{w^{-1}}} \mathfrak{L}^{(\beta)}_{\mathrm{wt}(K_-(P))}.$$

This conjecture proposes an alternative expansion of a Grothendieck polynomial in the Lascoux basis using increasing tableaux and their left keys, complementing the decreasing-tableau expansion proved in the paper. Its resolution is not specified in the supplied text.