Polo's Schubert-character positivity conjecture

Let $\alpha,\beta\in\mathbb{N}^m$, and let $\kappa_\alpha$ and $\kappa_\beta$ be Demazure characters. For a lower order ideal $\Gamma\subset\mathbb{N}^m$, let $\kappa_\Gamma$ be the corresponding Schubert character. Polo's character conjecture. There exist nonnegative integers $a_{\beta,\alpha}^{\Gamma}$ such that

$$\kappa_{\beta}\kappa_{\alpha}=\sum_{\Gamma}a_{\beta,\alpha}^{\Gamma}\kappa_{\Gamma}.$$

This is the character-level form of Polo's conjectured Schubert filtration. The paper notes that the corresponding Schubert-character statement has no computational test from the basis alone, while its weaker atom-positivity form is known in limited cases.