Lascoux positivity for characters of normal square root crystals

Let $m\in\mathbb Z_{>0}$, let $\mathcal C\subseteq\mathsf{RPD}_{m,n}$ be a $\sqrt{\mathfrak{gl}_n}$-crystal, and let $w\in S_n$. Write

$$\mathsf{ch}(\mathcal C_w)=\sum_{c\in\mathcal C_w}x^{\mathrm{wt}(c)}.$$

A polynomial is Lascoux positive if it is an $\mathbb N$-linear combination of Lascoux polynomials $\mathfrak L_\alpha$. Lascoux-positivity conjecture. For any $\sqrt{\mathfrak{gl}_n}$-crystal $\mathcal C\subseteq\mathsf{RPD}_{m,n}$, the polynomial $\mathsf{ch}(\mathcal C_w)$ is Lascoux positive.

This conjecture proposes a positivity property for the character of the subsets $\mathcal C_w$ associated with normal square root crystals. The supplied text gives no resolution or partial status beyond presenting it as a conjectural motivation.